id	sid	tid	token	lemma	pos
ejpam-1857	1	1	european	european	PROPN
ejpam-1857	1	2	journal	journal	PROPN
ejpam-1857	1	3	of	of	ADP
ejpam-1857	1	4	pure	pure	ADJ
ejpam-1857	1	5	and	and	CCONJ
ejpam-1857	1	6	applied	apply	VERB
ejpam-1857	1	7	mathematics	mathematic	NOUN
ejpam-1857	1	8	vol	vol	NOUN
ejpam-1857	1	9	.	.	PUNCT
ejpam-1857	2	1	7	7	NUM
ejpam-1857	2	2	,	,	PUNCT
ejpam-1857	2	3	no	no	INTJ
ejpam-1857	2	4	.	.	NOUN
ejpam-1857	2	5	2	2	NUM
ejpam-1857	2	6	,	,	PUNCT
ejpam-1857	2	7	2014	2014	NUM
ejpam-1857	2	8	,	,	PUNCT
ejpam-1857	2	9	131	131	NUM
ejpam-1857	2	10	-	-	SYM
ejpam-1857	2	11	139	139	NUM
ejpam-1857	2	12	issn	issn	PROPN
ejpam-1857	2	13	1307	1307	NUM
ejpam-1857	2	14	-	-	SYM
ejpam-1857	2	15	5543	5543	NUM
ejpam-1857	2	16	–	–	PUNCT
ejpam-1857	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-1857	2	18	elliptic	elliptic	ADJ
ejpam-1857	2	19	curves	curve	NOUN
ejpam-1857	2	20	and	and	CCONJ
ejpam-1857	2	21	pythagorean	pythagorean	PROPN
ejpam-1857	2	22	triples	triple	NOUN
ejpam-1857	2	23	farzali	farzali	ADJ
ejpam-1857	2	24	izadi	izadi	NOUN
ejpam-1857	2	25	,	,	PUNCT
ejpam-1857	2	26	kamran	kamran	PROPN
ejpam-1857	2	27	nabardi∗	nabardi∗	PROPN
ejpam-1857	2	28	department	department	PROPN
ejpam-1857	2	29	of	of	ADP
ejpam-1857	2	30	pure	pure	ADJ
ejpam-1857	2	31	mathematics	mathematic	NOUN
ejpam-1857	2	32	,	,	PUNCT
ejpam-1857	2	33	azarbaijan	azarbaijan	NOUN
ejpam-1857	2	34	shahid	shahid	PROPN
ejpam-1857	2	35	madani	madani	PROPN
ejpam-1857	2	36	university	university	PROPN
ejpam-1857	2	37	,	,	PUNCT
ejpam-1857	2	38	tabriz	tabriz	NOUN
ejpam-1857	2	39	,	,	PUNCT
ejpam-1857	2	40	iran	iran	PROPN
ejpam-1857	2	41	abstract	abstract	ADJ
ejpam-1857	2	42	.	.	PUNCT
ejpam-1857	3	1	the	the	DET
ejpam-1857	3	2	aim	aim	NOUN
ejpam-1857	3	3	of	of	ADP
ejpam-1857	3	4	this	this	DET
ejpam-1857	3	5	paper	paper	NOUN
ejpam-1857	3	6	is	be	AUX
ejpam-1857	3	7	to	to	PART
ejpam-1857	3	8	study	study	VERB
ejpam-1857	3	9	the	the	DET
ejpam-1857	3	10	family	family	NOUN
ejpam-1857	3	11	of	of	ADP
ejpam-1857	3	12	elliptic	elliptic	ADJ
ejpam-1857	3	13	curves	curve	NOUN
ejpam-1857	3	14	of	of	ADP
ejpam-1857	3	15	the	the	DET
ejpam-1857	3	16	form	form	NOUN
ejpam-1857	3	17	y2	y2	NOUN
ejpam-1857	3	18	=	=	SYM
ejpam-1857	3	19	x(x	x(x	PROPN
ejpam-1857	4	1	−	−	PROPN
ejpam-1857	4	2	a2)(x	a2)(x	PROPN
ejpam-1857	4	3	−	−	PROPN
ejpam-1857	4	4	b2	b2	NOUN
ejpam-1857	4	5	)	)	PUNCT
ejpam-1857	4	6	,	,	PUNCT
ejpam-1857	4	7	where	where	SCONJ
ejpam-1857	4	8	(	(	PUNCT
ejpam-1857	4	9	a	a	DET
ejpam-1857	4	10	,	,	PUNCT
ejpam-1857	4	11	b	b	NOUN
ejpam-1857	4	12	,	,	PUNCT
ejpam-1857	4	13	c	c	NOUN
ejpam-1857	4	14	)	)	PUNCT
ejpam-1857	4	15	is	be	AUX
ejpam-1857	4	16	a	a	DET
ejpam-1857	4	17	primitive	primitive	ADJ
ejpam-1857	4	18	pythagorean	pythagorean	NOUN
ejpam-1857	4	19	triple	triple	NOUN
ejpam-1857	4	20	.	.	PUNCT
ejpam-1857	5	1	first	first	ADV
ejpam-1857	5	2	we	we	PRON
ejpam-1857	5	3	show	show	VERB
ejpam-1857	5	4	that	that	SCONJ
ejpam-1857	5	5	the	the	DET
ejpam-1857	5	6	rank	rank	NOUN
ejpam-1857	5	7	is	be	AUX
ejpam-1857	5	8	positive	positive	ADJ
ejpam-1857	5	9	.	.	PUNCT
ejpam-1857	6	1	then	then	ADV
ejpam-1857	6	2	we	we	PRON
ejpam-1857	6	3	construct	construct	VERB
ejpam-1857	6	4	some	some	DET
ejpam-1857	6	5	subfamilies	subfamily	NOUN
ejpam-1857	6	6	with	with	ADP
ejpam-1857	6	7	rank	rank	PROPN
ejpam-1857	6	8	≥	≥	NOUN
ejpam-1857	6	9	2	2	NUM
ejpam-1857	6	10	by	by	ADP
ejpam-1857	6	11	different	different	ADJ
ejpam-1857	6	12	methods	method	NOUN
ejpam-1857	6	13	.	.	PUNCT
ejpam-1857	7	1	2010	2010	NUM
ejpam-1857	7	2	mathematics	mathematic	NOUN
ejpam-1857	7	3	subject	subject	NOUN
ejpam-1857	7	4	classifications	classification	NOUN
ejpam-1857	7	5	:	:	PUNCT
ejpam-1857	7	6	11g05	11g05	NUM
ejpam-1857	7	7	,	,	PUNCT
ejpam-1857	7	8	14h52	14h52	NUM
ejpam-1857	7	9	,	,	PUNCT
ejpam-1857	7	10	14g05	14g05	NUM
ejpam-1857	7	11	key	key	ADJ
ejpam-1857	7	12	words	word	NOUN
ejpam-1857	7	13	and	and	CCONJ
ejpam-1857	7	14	phrases	phrase	NOUN
ejpam-1857	7	15	:	:	PUNCT
ejpam-1857	7	16	elliptic	elliptic	ADJ
ejpam-1857	7	17	curves	curve	NOUN
ejpam-1857	7	18	,	,	PUNCT
ejpam-1857	7	19	rank	rank	NOUN
ejpam-1857	7	20	,	,	PUNCT
ejpam-1857	7	21	pythagorean	pythagorean	PROPN
ejpam-1857	7	22	triples	triple	NOUN
ejpam-1857	7	23	1	1	NUM
ejpam-1857	7	24	.	.	PUNCT
ejpam-1857	7	25	introduction	introduction	NOUN
ejpam-1857	7	26	an	an	DET
ejpam-1857	7	27	elliptic	elliptic	ADJ
ejpam-1857	7	28	curve	curve	NOUN
ejpam-1857	7	29	e	e	NOUN
ejpam-1857	7	30	over	over	ADP
ejpam-1857	7	31	the	the	DET
ejpam-1857	7	32	rational	rational	ADJ
ejpam-1857	7	33	field	field	NOUN
ejpam-1857	7	34	q	q	PUNCT
ejpam-1857	7	35	is	be	AUX
ejpam-1857	7	36	a	a	DET
ejpam-1857	7	37	curve	curve	NOUN
ejpam-1857	7	38	that	that	PRON
ejpam-1857	7	39	is	be	AUX
ejpam-1857	7	40	given	give	VERB
ejpam-1857	7	41	by	by	ADP
ejpam-1857	7	42	y	y	PROPN
ejpam-1857	7	43	2	2	NUM
ejpam-1857	7	44	=	=	SYM
ejpam-1857	7	45	x	x	SYM
ejpam-1857	7	46	3	3	NUM
ejpam-1857	7	47	+	+	CCONJ
ejpam-1857	7	48	ax	ax	NOUN
ejpam-1857	7	49	2	2	NUM
ejpam-1857	7	50	+	+	CCONJ
ejpam-1857	7	51	bx	bx	NOUN
ejpam-1857	7	52	+	+	CCONJ
ejpam-1857	7	53	c	c	X
ejpam-1857	7	54	,	,	PUNCT
ejpam-1857	7	55	a	a	DET
ejpam-1857	7	56	,	,	PUNCT
ejpam-1857	7	57	b	b	NOUN
ejpam-1857	7	58	,	,	PUNCT
ejpam-1857	7	59	c	c	PROPN
ejpam-1857	7	60	∈q	∈q	NOUN
ejpam-1857	7	61	,	,	PUNCT
ejpam-1857	7	62	(	(	PUNCT
ejpam-1857	7	63	1	1	NUM
ejpam-1857	7	64	)	)	PUNCT
ejpam-1857	7	65	with	with	ADP
ejpam-1857	7	66	the	the	DET
ejpam-1857	7	67	condition	condition	NOUN
ejpam-1857	7	68	that	that	SCONJ
ejpam-1857	7	69	the	the	DET
ejpam-1857	7	70	polynomial	polynomial	ADJ
ejpam-1857	7	71	x	x	SYM
ejpam-1857	7	72	3	3	NUM
ejpam-1857	7	73	+	+	NOUN
ejpam-1857	7	74	ax	ax	NOUN
ejpam-1857	7	75	2	2	NUM
ejpam-1857	7	76	+	+	CCONJ
ejpam-1857	7	77	bx	bx	PRON
ejpam-1857	7	78	+	+	CCONJ
ejpam-1857	7	79	c	c	NOUN
ejpam-1857	7	80	has	have	AUX
ejpam-1857	7	81	no	no	DET
ejpam-1857	7	82	multiple	multiple	ADJ
ejpam-1857	7	83	zeroes	zero	NOUN
ejpam-1857	7	84	.	.	PUNCT
ejpam-1857	8	1	mordell	mordell	PROPN
ejpam-1857	8	2	proved	prove	VERB
ejpam-1857	8	3	that	that	SCONJ
ejpam-1857	8	4	on	on	ADP
ejpam-1857	8	5	an	an	DET
ejpam-1857	8	6	elliptic	elliptic	ADJ
ejpam-1857	8	7	curve	curve	NOUN
ejpam-1857	8	8	over	over	ADP
ejpam-1857	8	9	q	q	NOUN
ejpam-1857	8	10	,	,	PUNCT
ejpam-1857	8	11	the	the	DET
ejpam-1857	8	12	rational	rational	ADJ
ejpam-1857	8	13	points	point	NOUN
ejpam-1857	8	14	form	form	VERB
ejpam-1857	8	15	a	a	DET
ejpam-1857	8	16	finitely	finitely	ADV
ejpam-1857	8	17	generated	generate	VERB
ejpam-1857	8	18	abelian	abelian	ADJ
ejpam-1857	8	19	group	group	NOUN
ejpam-1857	8	20	which	which	PRON
ejpam-1857	8	21	is	be	AUX
ejpam-1857	8	22	denoted	denote	VERB
ejpam-1857	8	23	by	by	ADP
ejpam-1857	8	24	e(q	e(q	NOUN
ejpam-1857	8	25	)	)	PUNCT
ejpam-1857	9	1	[	[	X
ejpam-1857	9	2	2	2	NUM
ejpam-1857	9	3	]	]	PUNCT
ejpam-1857	9	4	.	.	PUNCT
ejpam-1857	10	1	here	here	ADV
ejpam-1857	10	2	we	we	PRON
ejpam-1857	10	3	can	can	AUX
ejpam-1857	10	4	apply	apply	VERB
ejpam-1857	10	5	the	the	DET
ejpam-1857	10	6	structure	structure	NOUN
ejpam-1857	10	7	theorem	theorem	NOUN
ejpam-1857	10	8	for	for	ADP
ejpam-1857	10	9	the	the	DET
ejpam-1857	10	10	finitely	finitely	ADV
ejpam-1857	10	11	generated	generate	VERB
ejpam-1857	10	12	abelian	abelian	ADJ
ejpam-1857	10	13	groups	group	NOUN
ejpam-1857	10	14	to	to	PART
ejpam-1857	10	15	e(q	e(q	NOUN
ejpam-1857	10	16	)	)	PUNCT
ejpam-1857	10	17	to	to	PART
ejpam-1857	10	18	obtain	obtain	VERB
ejpam-1857	10	19	a	a	DET
ejpam-1857	10	20	decomposition	decomposition	NOUN
ejpam-1857	10	21	of	of	ADP
ejpam-1857	10	22	e(q	e(q	NOUN
ejpam-1857	10	23	)	)	PUNCT
ejpam-1857	10	24	'	'	PUNCT
ejpam-1857	10	25	zr	zr	NUM
ejpam-1857	10	26	×	×	PROPN
ejpam-1857	10	27	e(q)tors	e(q)tor	NOUN
ejpam-1857	10	28	,	,	PUNCT
ejpam-1857	10	29	where	where	SCONJ
ejpam-1857	10	30	r	r	NOUN
ejpam-1857	10	31	is	be	AUX
ejpam-1857	10	32	an	an	DET
ejpam-1857	10	33	integer	integer	NOUN
ejpam-1857	10	34	called	call	VERB
ejpam-1857	10	35	the	the	DET
ejpam-1857	10	36	rank	rank	NOUN
ejpam-1857	10	37	of	of	ADP
ejpam-1857	10	38	e	e	PROPN
ejpam-1857	10	39	and	and	CCONJ
ejpam-1857	10	40	e(q)tors	e(q)tor	NOUN
ejpam-1857	10	41	is	be	AUX
ejpam-1857	10	42	the	the	DET
ejpam-1857	10	43	finite	finite	PROPN
ejpam-1857	10	44	abelian	abelian	PROPN
ejpam-1857	10	45	group	group	PROPN
ejpam-1857	10	46	consisting	consist	VERB
ejpam-1857	10	47	of	of	ADP
ejpam-1857	10	48	all	all	DET
ejpam-1857	10	49	elements	element	NOUN
ejpam-1857	10	50	of	of	ADP
ejpam-1857	10	51	finite	finite	ADJ
ejpam-1857	10	52	order	order	NOUN
ejpam-1857	10	53	in	in	ADP
ejpam-1857	10	54	e(q	e(q	NOUN
ejpam-1857	10	55	)	)	PUNCT
ejpam-1857	10	56	.	.	PUNCT
ejpam-1857	11	1	in	in	ADP
ejpam-1857	11	2	1976	1976	NUM
ejpam-1857	11	3	,	,	PUNCT
ejpam-1857	11	4	barry	barry	PROPN
ejpam-1857	11	5	mazur	mazur	PROPN
ejpam-1857	11	6	proved	prove	VERB
ejpam-1857	11	7	the	the	DET
ejpam-1857	11	8	following	follow	VERB
ejpam-1857	11	9	seminal	seminal	ADJ
ejpam-1857	11	10	result	result	NOUN
ejpam-1857	11	11	[	[	X
ejpam-1857	11	12	4	4	NUM
ejpam-1857	11	13	]	]	PUNCT
ejpam-1857	11	14	.	.	PUNCT
ejpam-1857	12	1	the	the	DET
ejpam-1857	12	2	torsion	torsion	NOUN
ejpam-1857	12	3	group	group	NOUN
ejpam-1857	12	4	e(q)tors	e(q)tor	NOUN
ejpam-1857	12	5	of	of	ADP
ejpam-1857	12	6	any	any	DET
ejpam-1857	12	7	elliptic	elliptic	ADJ
ejpam-1857	12	8	curve	curve	NOUN
ejpam-1857	12	9	e	e	NOUN
ejpam-1857	12	10	over	over	ADP
ejpam-1857	12	11	q	q	NOUN
ejpam-1857	12	12	is	be	AUX
ejpam-1857	12	13	one	one	NUM
ejpam-1857	12	14	of	of	ADP
ejpam-1857	12	15	the	the	DET
ejpam-1857	12	16	following	follow	VERB
ejpam-1857	12	17	15	15	NUM
ejpam-1857	12	18	types	type	NOUN
ejpam-1857	12	19	.	.	PUNCT
ejpam-1857	13	1	moreover	moreover	ADV
ejpam-1857	13	2	,	,	PUNCT
ejpam-1857	13	3	each	each	PRON
ejpam-1857	13	4	of	of	ADP
ejpam-1857	13	5	these	these	DET
ejpam-1857	13	6	cases	case	NOUN
ejpam-1857	13	7	occurs	occur	VERB
ejpam-1857	13	8	for	for	ADP
ejpam-1857	13	9	infinitely	infinitely	ADV
ejpam-1857	13	10	many	many	ADJ
ejpam-1857	13	11	curves	curve	NOUN
ejpam-1857	13	12	e	e	NOUN
ejpam-1857	13	13	over	over	ADP
ejpam-1857	13	14	q.	q.	PROPN
ejpam-1857	13	15	¨	¨	NOUN
ejpam-1857	14	1	z	z	X
ejpam-1857	14	2	/	/	SYM
ejpam-1857	14	3	mz	mz	PROPN
ejpam-1857	14	4	m=	m=	X
ejpam-1857	14	5	1,2	1,2	NUM
ejpam-1857	14	6	,	,	PUNCT
ejpam-1857	14	7	3	3	NUM
ejpam-1857	14	8	,	,	PUNCT
ejpam-1857	14	9	.	.	PUNCT
ejpam-1857	14	10	.	.	PUNCT
ejpam-1857	15	1	.	.	PUNCT
ejpam-1857	16	1	,	,	PUNCT
ejpam-1857	16	2	10	10	NUM
ejpam-1857	16	3	,	,	PUNCT
ejpam-1857	16	4	12	12	NUM
ejpam-1857	16	5	,	,	PUNCT
ejpam-1857	16	6	z/2z×z	z/2z×z	NUM
ejpam-1857	16	7	/	/	SYM
ejpam-1857	16	8	mz	mz	PROPN
ejpam-1857	16	9	m=	m=	X
ejpam-1857	16	10	2,4	2,4	NUM
ejpam-1857	16	11	,	,	PUNCT
ejpam-1857	16	12	6,8	6,8	NUM
ejpam-1857	16	13	.	.	PUNCT
ejpam-1857	17	1	(	(	PUNCT
ejpam-1857	17	2	2	2	X
ejpam-1857	17	3	)	)	PUNCT
ejpam-1857	17	4	this	this	PRON
ejpam-1857	17	5	shows	show	VERB
ejpam-1857	17	6	that	that	SCONJ
ejpam-1857	17	7	e(q	e(q	NOUN
ejpam-1857	17	8	)	)	PUNCT
ejpam-1857	17	9	can	can	AUX
ejpam-1857	17	10	not	not	PART
ejpam-1857	17	11	contain	contain	VERB
ejpam-1857	17	12	a	a	DET
ejpam-1857	17	13	point	point	NOUN
ejpam-1857	17	14	of	of	ADP
ejpam-1857	17	15	order	order	NOUN
ejpam-1857	17	16	11	11	NUM
ejpam-1857	17	17	,	,	PUNCT
ejpam-1857	17	18	nor	nor	CCONJ
ejpam-1857	17	19	of	of	ADP
ejpam-1857	17	20	any	any	DET
ejpam-1857	17	21	order	order	NOUN
ejpam-1857	17	22	n≥	n≥	NOUN
ejpam-1857	17	23	13	13	NUM
ejpam-1857	17	24	.	.	PUNCT
ejpam-1857	18	1	∗corresponding	∗corresponde	VERB
ejpam-1857	18	2	author	author	NOUN
ejpam-1857	18	3	.	.	PUNCT
ejpam-1857	19	1	email	email	NOUN
ejpam-1857	19	2	addresses	address	NOUN
ejpam-1857	19	3	:	:	PUNCT
ejpam-1857	19	4	farzali.izadi@azaruniv.edu	farzali.izadi@azaruniv.edu	NOUN
ejpam-1857	19	5	(	(	PUNCT
ejpam-1857	19	6	f.	f.	PROPN
ejpam-1857	19	7	izadi	izadi	PROPN
ejpam-1857	19	8	)	)	PUNCT
ejpam-1857	19	9	,	,	PUNCT
ejpam-1857	19	10	nabardi@azaruniv.edu	nabardi@azaruniv.edu	PROPN
ejpam-1857	19	11	(	(	PUNCT
ejpam-1857	19	12	k.	k.	PROPN
ejpam-1857	19	13	nabardi	nabardi	PROPN
ejpam-1857	19	14	)	)	PUNCT
ejpam-1857	19	15	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-1857	20	1	131	131	NUM
ejpam-1857	21	1	c	c	X
ejpam-1857	21	2	©	©	PROPN
ejpam-1857	21	3	2014	2014	NUM
ejpam-1857	21	4	ejpam	ejpam	NOUN
ejpam-1857	21	5	all	all	DET
ejpam-1857	21	6	rights	right	NOUN
ejpam-1857	21	7	reserved	reserve	VERB
ejpam-1857	21	8	.	.	PUNCT
ejpam-1857	22	1	f.	f.	PROPN
ejpam-1857	22	2	izadi	izadi	PROPN
ejpam-1857	22	3	,	,	PUNCT
ejpam-1857	22	4	k.	k.	PROPN
ejpam-1857	22	5	nabardi	nabardi	PROPN
ejpam-1857	22	6	/	/	SYM
ejpam-1857	22	7	eur	eur	PROPN
ejpam-1857	22	8	.	.	PUNCT
ejpam-1857	23	1	j.	j.	PROPN
ejpam-1857	23	2	pure	pure	PROPN
ejpam-1857	23	3	appl	appl	PROPN
ejpam-1857	23	4	.	.	PROPN
ejpam-1857	23	5	math	math	PROPN
ejpam-1857	23	6	,	,	PUNCT
ejpam-1857	23	7	7	7	NUM
ejpam-1857	23	8	(	(	PUNCT
ejpam-1857	23	9	2014	2014	NUM
ejpam-1857	23	10	)	)	PUNCT
ejpam-1857	23	11	,	,	PUNCT
ejpam-1857	23	12	131	131	NUM
ejpam-1857	23	13	-	-	SYM
ejpam-1857	23	14	139	139	NUM
ejpam-1857	23	15	132	132	NUM
ejpam-1857	23	16	on	on	ADP
ejpam-1857	23	17	the	the	DET
ejpam-1857	23	18	other	other	ADJ
ejpam-1857	23	19	hand	hand	NOUN
ejpam-1857	23	20	,	,	PUNCT
ejpam-1857	23	21	it	it	PRON
ejpam-1857	23	22	is	be	AUX
ejpam-1857	23	23	not	not	PART
ejpam-1857	23	24	known	know	VERB
ejpam-1857	23	25	which	which	DET
ejpam-1857	23	26	values	value	NOUN
ejpam-1857	23	27	of	of	ADP
ejpam-1857	23	28	r	r	NOUN
ejpam-1857	23	29	are	be	AUX
ejpam-1857	23	30	possible	possible	ADJ
ejpam-1857	23	31	.	.	PUNCT
ejpam-1857	24	1	the	the	DET
ejpam-1857	24	2	current	current	ADJ
ejpam-1857	24	3	record	record	NOUN
ejpam-1857	24	4	is	be	AUX
ejpam-1857	24	5	an	an	DET
ejpam-1857	24	6	example	example	NOUN
ejpam-1857	24	7	of	of	ADP
ejpam-1857	24	8	elliptic	elliptic	ADJ
ejpam-1857	24	9	curve	curve	NOUN
ejpam-1857	24	10	over	over	ADP
ejpam-1857	24	11	q	q	NOUN
ejpam-1857	24	12	with	with	ADP
ejpam-1857	24	13	r	r	PROPN
ejpam-1857	24	14	≥	≥	NUM
ejpam-1857	24	15	28	28	NUM
ejpam-1857	24	16	found	find	VERB
ejpam-1857	24	17	by	by	ADP
ejpam-1857	24	18	elkies	elkie	NOUN
ejpam-1857	24	19	in	in	ADP
ejpam-1857	24	20	may	may	PROPN
ejpam-1857	24	21	2006	2006	NUM
ejpam-1857	24	22	.	.	PUNCT
ejpam-1857	25	1	in	in	ADP
ejpam-1857	25	2	this	this	DET
ejpam-1857	25	3	paper	paper	NOUN
ejpam-1857	25	4	,	,	PUNCT
ejpam-1857	25	5	we	we	PRON
ejpam-1857	25	6	first	first	ADV
ejpam-1857	25	7	introduce	introduce	VERB
ejpam-1857	25	8	a	a	DET
ejpam-1857	25	9	family	family	NOUN
ejpam-1857	25	10	of	of	ADP
ejpam-1857	25	11	elliptic	elliptic	ADJ
ejpam-1857	25	12	curves	curve	NOUN
ejpam-1857	25	13	over	over	ADP
ejpam-1857	25	14	q	q	NOUN
ejpam-1857	25	15	of	of	ADP
ejpam-1857	25	16	the	the	DET
ejpam-1857	25	17	form	form	NOUN
ejpam-1857	25	18	y2	y2	NOUN
ejpam-1857	25	19	=	=	SYM
ejpam-1857	25	20	x(x	x(x	PROPN
ejpam-1857	26	1	−	−	PROPN
ejpam-1857	26	2	a2)(x	a2)(x	PROPN
ejpam-1857	26	3	−	−	PROPN
ejpam-1857	26	4	b2	b2	NOUN
ejpam-1857	26	5	)	)	PUNCT
ejpam-1857	26	6	,	,	PUNCT
ejpam-1857	26	7	where	where	SCONJ
ejpam-1857	26	8	,	,	PUNCT
ejpam-1857	26	9	(	(	PUNCT
ejpam-1857	26	10	a	a	DET
ejpam-1857	26	11	,	,	PUNCT
ejpam-1857	26	12	b	b	NOUN
ejpam-1857	26	13	,	,	PUNCT
ejpam-1857	26	14	c	c	NOUN
ejpam-1857	26	15	)	)	PUNCT
ejpam-1857	26	16	is	be	AUX
ejpam-1857	26	17	a	a	DET
ejpam-1857	26	18	pythagorean	pythagorean	PROPN
ejpam-1857	26	19	triple	triple	NOUN
ejpam-1857	26	20	and	and	CCONJ
ejpam-1857	26	21	show	show	VERB
ejpam-1857	26	22	that	that	SCONJ
ejpam-1857	26	23	they	they	PRON
ejpam-1857	26	24	have	have	VERB
ejpam-1857	26	25	positive	positive	ADJ
ejpam-1857	26	26	ranks	rank	NOUN
ejpam-1857	26	27	.	.	PUNCT
ejpam-1857	27	1	in	in	ADP
ejpam-1857	27	2	section	section	NOUN
ejpam-1857	27	3	2	2	NUM
ejpam-1857	27	4	,	,	PUNCT
ejpam-1857	27	5	we	we	PRON
ejpam-1857	27	6	briefly	briefly	ADV
ejpam-1857	27	7	describe	describe	VERB
ejpam-1857	27	8	the	the	DET
ejpam-1857	27	9	construction	construction	NOUN
ejpam-1857	27	10	of	of	ADP
ejpam-1857	27	11	this	this	DET
ejpam-1857	27	12	family	family	NOUN
ejpam-1857	27	13	and	and	CCONJ
ejpam-1857	27	14	show	show	VERB
ejpam-1857	27	15	that	that	SCONJ
ejpam-1857	27	16	its	its	PRON
ejpam-1857	27	17	the	the	DET
ejpam-1857	27	18	torsion	torsion	NOUN
ejpam-1857	27	19	group	group	NOUN
ejpam-1857	27	20	is	be	AUX
ejpam-1857	27	21	z/2z×z/2z	z/2z×z/2z	PROPN
ejpam-1857	27	22	.	.	PUNCT
ejpam-1857	28	1	then	then	ADV
ejpam-1857	28	2	we	we	PRON
ejpam-1857	28	3	prove	prove	VERB
ejpam-1857	28	4	that	that	SCONJ
ejpam-1857	28	5	it	it	PRON
ejpam-1857	28	6	has	have	VERB
ejpam-1857	28	7	positive	positive	ADJ
ejpam-1857	28	8	rank	rank	NOUN
ejpam-1857	28	9	.	.	PUNCT
ejpam-1857	29	1	finally	finally	ADV
ejpam-1857	29	2	by	by	ADP
ejpam-1857	29	3	using	use	VERB
ejpam-1857	29	4	by	by	ADP
ejpam-1857	29	5	mrank	mrank	ADJ
ejpam-1857	29	6	program	program	NOUN
ejpam-1857	29	7	[	[	X
ejpam-1857	29	8	1	1	NUM
ejpam-1857	29	9	]	]	PUNCT
ejpam-1857	29	10	,	,	PUNCT
ejpam-1857	29	11	we	we	PRON
ejpam-1857	29	12	can	can	AUX
ejpam-1857	29	13	find	find	VERB
ejpam-1857	29	14	some	some	DET
ejpam-1857	29	15	curves	curve	NOUN
ejpam-1857	29	16	with	with	ADP
ejpam-1857	29	17	rank	rank	NOUN
ejpam-1857	29	18	5	5	NUM
ejpam-1857	29	19	.	.	PUNCT
ejpam-1857	30	1	in	in	ADP
ejpam-1857	30	2	section	section	NOUN
ejpam-1857	30	3	3	3	NUM
ejpam-1857	30	4	,	,	PUNCT
ejpam-1857	30	5	we	we	PRON
ejpam-1857	30	6	describe	describe	VERB
ejpam-1857	30	7	a	a	DET
ejpam-1857	30	8	method	method	NOUN
ejpam-1857	30	9	to	to	PART
ejpam-1857	30	10	find	find	VERB
ejpam-1857	30	11	a	a	DET
ejpam-1857	30	12	subfamily	subfamily	NOUN
ejpam-1857	30	13	with	with	ADP
ejpam-1857	30	14	rank	rank	PROPN
ejpam-1857	30	15	≥	≥	NOUN
ejpam-1857	30	16	2	2	NUM
ejpam-1857	30	17	.	.	PUNCT
ejpam-1857	31	1	the	the	DET
ejpam-1857	31	2	resulting	result	VERB
ejpam-1857	31	3	subfamily	subfamily	ADV
ejpam-1857	31	4	corresponds	correspond	VERB
ejpam-1857	31	5	to	to	ADP
ejpam-1857	31	6	the	the	DET
ejpam-1857	31	7	points	point	NOUN
ejpam-1857	31	8	of	of	ADP
ejpam-1857	31	9	a	a	DET
ejpam-1857	31	10	specific	specific	ADJ
ejpam-1857	31	11	elliptic	elliptic	ADJ
ejpam-1857	31	12	curve	curve	NOUN
ejpam-1857	31	13	having	have	VERB
ejpam-1857	31	14	positive	positive	ADJ
ejpam-1857	31	15	rank	rank	NOUN
ejpam-1857	31	16	too	too	ADV
ejpam-1857	31	17	.	.	PUNCT
ejpam-1857	32	1	finally	finally	ADV
ejpam-1857	32	2	in	in	ADP
ejpam-1857	32	3	section	section	NOUN
ejpam-1857	32	4	4	4	NUM
ejpam-1857	32	5	,	,	PUNCT
ejpam-1857	32	6	by	by	ADP
ejpam-1857	32	7	letting	let	VERB
ejpam-1857	32	8	a	a	DET
ejpam-1857	32	9	=	=	PUNCT
ejpam-1857	32	10	t2	t2	NOUN
ejpam-1857	32	11	−	−	PROPN
ejpam-1857	32	12	1	1	NUM
ejpam-1857	32	13	,	,	PUNCT
ejpam-1857	32	14	b	b	NOUN
ejpam-1857	32	15	=	=	SYM
ejpam-1857	32	16	2	2	NUM
ejpam-1857	32	17	t	t	NOUN
ejpam-1857	32	18	,	,	PUNCT
ejpam-1857	32	19	and	and	CCONJ
ejpam-1857	32	20	c	c	X
ejpam-1857	32	21	=	=	SYM
ejpam-1857	32	22	t2	t2	PROPN
ejpam-1857	32	23	+	+	CCONJ
ejpam-1857	32	24	1	1	NUM
ejpam-1857	32	25	as	as	ADP
ejpam-1857	32	26	functions	function	NOUN
ejpam-1857	32	27	of	of	ADP
ejpam-1857	32	28	the	the	DET
ejpam-1857	32	29	rational	rational	ADJ
ejpam-1857	32	30	parameter	parameter	NOUN
ejpam-1857	32	31	t	t	PROPN
ejpam-1857	32	32	,	,	PUNCT
ejpam-1857	32	33	we	we	PRON
ejpam-1857	32	34	study	study	VERB
ejpam-1857	32	35	the	the	DET
ejpam-1857	32	36	family	family	NOUN
ejpam-1857	32	37	as	as	ADP
ejpam-1857	32	38	a	a	DET
ejpam-1857	32	39	parameter	parameter	NOUN
ejpam-1857	32	40	family	family	NOUN
ejpam-1857	32	41	and	and	CCONJ
ejpam-1857	32	42	show	show	VERB
ejpam-1857	32	43	that	that	SCONJ
ejpam-1857	32	44	it	it	PRON
ejpam-1857	32	45	has	have	VERB
ejpam-1857	32	46	a	a	DET
ejpam-1857	32	47	subfamily	subfamily	NOUN
ejpam-1857	32	48	of	of	ADP
ejpam-1857	32	49	rank	rank	NOUN
ejpam-1857	32	50	≥	≥	NOUN
ejpam-1857	32	51	2	2	NUM
ejpam-1857	32	52	.	.	PUNCT
ejpam-1857	33	1	by	by	ADP
ejpam-1857	33	2	this	this	DET
ejpam-1857	33	3	method	method	NOUN
ejpam-1857	33	4	we	we	PRON
ejpam-1857	33	5	can	can	AUX
ejpam-1857	33	6	find	find	VERB
ejpam-1857	33	7	4	4	NUM
ejpam-1857	33	8	curves	curve	NOUN
ejpam-1857	33	9	of	of	ADP
ejpam-1857	33	10	rank	rank	NOUN
ejpam-1857	33	11	equal	equal	ADJ
ejpam-1857	33	12	to	to	ADP
ejpam-1857	33	13	6	6	NUM
ejpam-1857	33	14	.	.	PUNCT
ejpam-1857	33	15	remark	remark	NOUN
ejpam-1857	33	16	1	1	NUM
ejpam-1857	33	17	.	.	PUNCT
ejpam-1857	34	1	if	if	SCONJ
ejpam-1857	34	2	s	s	NOUN
ejpam-1857	34	3	is	be	AUX
ejpam-1857	34	4	any	any	DET
ejpam-1857	34	5	nonzero	nonzero	ADJ
ejpam-1857	34	6	rational	rational	ADJ
ejpam-1857	34	7	number	number	NOUN
ejpam-1857	34	8	,	,	PUNCT
ejpam-1857	34	9	then	then	ADV
ejpam-1857	34	10	replacing	replace	VERB
ejpam-1857	34	11	(	(	PUNCT
ejpam-1857	34	12	a	a	DET
ejpam-1857	34	13	,	,	PUNCT
ejpam-1857	34	14	b	b	NOUN
ejpam-1857	34	15	,	,	PUNCT
ejpam-1857	34	16	c	c	NOUN
ejpam-1857	34	17	)	)	PUNCT
ejpam-1857	34	18	by	by	ADP
ejpam-1857	34	19	(	(	PUNCT
ejpam-1857	34	20	sa	sa	PROPN
ejpam-1857	34	21	,	,	PUNCT
ejpam-1857	34	22	sb	sb	PROPN
ejpam-1857	34	23	,	,	PUNCT
ejpam-1857	34	24	sc	sc	PROPN
ejpam-1857	34	25	)	)	PUNCT
ejpam-1857	34	26	,	,	PUNCT
ejpam-1857	34	27	one	one	PRON
ejpam-1857	34	28	has	have	VERB
ejpam-1857	34	29	(	(	PUNCT
ejpam-1857	34	30	sa)2	sa)2	NOUN
ejpam-1857	34	31	+	+	CCONJ
ejpam-1857	34	32	(	(	PUNCT
ejpam-1857	34	33	sb)2	sb)2	PROPN
ejpam-1857	34	34	=	=	SYM
ejpam-1857	34	35	(	(	PUNCT
ejpam-1857	34	36	sc)2	sc)2	NOUN
ejpam-1857	34	37	(	(	PUNCT
ejpam-1857	34	38	but	but	CCONJ
ejpam-1857	34	39	possibly	possibly	ADV
ejpam-1857	34	40	these	these	DET
ejpam-1857	34	41	numbers	number	NOUN
ejpam-1857	34	42	are	be	AUX
ejpam-1857	34	43	rational	rational	ADJ
ejpam-1857	34	44	rather	rather	ADV
ejpam-1857	34	45	than	than	ADP
ejpam-1857	34	46	integral	integral	ADJ
ejpam-1857	34	47	)	)	PUNCT
ejpam-1857	34	48	,	,	PUNCT
ejpam-1857	34	49	and	and	CCONJ
ejpam-1857	34	50	the	the	DET
ejpam-1857	34	51	corresponding	corresponding	ADJ
ejpam-1857	34	52	elliptic	elliptic	ADJ
ejpam-1857	34	53	curve	curve	NOUN
ejpam-1857	34	54	y2	y2	PROPN
ejpam-1857	34	55	=	=	SYM
ejpam-1857	34	56	x(x	x(x	PROPN
ejpam-1857	35	1	−	−	PROPN
ejpam-1857	35	2	(	(	PUNCT
ejpam-1857	35	3	sa)2)(x	sa)2)(x	NOUN
ejpam-1857	35	4	−	−	PROPN
ejpam-1857	35	5	(	(	PUNCT
ejpam-1857	35	6	sb)2	sb)2	PROPN
ejpam-1857	35	7	)	)	PUNCT
ejpam-1857	35	8	,	,	PUNCT
ejpam-1857	35	9	is	be	AUX
ejpam-1857	35	10	over	over	ADP
ejpam-1857	35	11	q	q	NOUN
ejpam-1857	35	12	isomorphic	isomorphic	ADJ
ejpam-1857	35	13	to	to	ADP
ejpam-1857	35	14	the	the	DET
ejpam-1857	35	15	original	original	ADJ
ejpam-1857	35	16	one	one	NUM
ejpam-1857	35	17	.	.	PUNCT
ejpam-1857	36	1	the	the	DET
ejpam-1857	36	2	isomorphism	isomorphism	NOUN
ejpam-1857	36	3	is	be	AUX
ejpam-1857	36	4	given	give	VERB
ejpam-1857	36	5	by	by	ADP
ejpam-1857	36	6	(	(	PUNCT
ejpam-1857	36	7	x	x	INTJ
ejpam-1857	36	8	,	,	PUNCT
ejpam-1857	36	9	y	y	PROPN
ejpam-1857	36	10	)	)	PUNCT
ejpam-1857	36	11	→	→	PUNCT
ejpam-1857	36	12	(	(	PUNCT
ejpam-1857	36	13	s2	s2	NOUN
ejpam-1857	36	14	x	x	X
ejpam-1857	36	15	,	,	PUNCT
ejpam-1857	36	16	s3	s3	PROPN
ejpam-1857	36	17	y	y	PROPN
ejpam-1857	36	18	)	)	PUNCT
ejpam-1857	36	19	.	.	PUNCT
ejpam-1857	37	1	in	in	ADP
ejpam-1857	37	2	particular	particular	ADJ
ejpam-1857	37	3	this	this	PRON
ejpam-1857	37	4	implies	imply	VERB
ejpam-1857	37	5	that	that	SCONJ
ejpam-1857	37	6	it	it	PRON
ejpam-1857	37	7	is	be	AUX
ejpam-1857	37	8	not	not	PART
ejpam-1857	37	9	necessary	necessary	ADJ
ejpam-1857	37	10	to	to	PART
ejpam-1857	37	11	demand	demand	VERB
ejpam-1857	37	12	that	that	SCONJ
ejpam-1857	37	13	the	the	DET
ejpam-1857	37	14	pythagorean	pythagorean	PROPN
ejpam-1857	37	15	triple	triple	NOUN
ejpam-1857	37	16	is	be	AUX
ejpam-1857	37	17	primitive	primitive	ADJ
ejpam-1857	37	18	.	.	PUNCT
ejpam-1857	38	1	whenever	whenever	SCONJ
ejpam-1857	38	2	it	it	PRON
ejpam-1857	38	3	is	be	AUX
ejpam-1857	38	4	convenient	convenient	ADJ
ejpam-1857	38	5	in	in	ADP
ejpam-1857	38	6	some	some	DET
ejpam-1857	38	7	proof	proof	NOUN
ejpam-1857	38	8	,	,	PUNCT
ejpam-1857	38	9	we	we	PRON
ejpam-1857	38	10	can	can	AUX
ejpam-1857	38	11	assume	assume	VERB
ejpam-1857	38	12	the	the	DET
ejpam-1857	38	13	triple	triple	ADJ
ejpam-1857	38	14	to	to	PART
ejpam-1857	38	15	be	be	AUX
ejpam-1857	38	16	primitive	primitive	ADJ
ejpam-1857	38	17	,	,	PUNCT
ejpam-1857	38	18	without	without	ADP
ejpam-1857	38	19	loss	loss	NOUN
ejpam-1857	38	20	of	of	ADP
ejpam-1857	38	21	generality	generality	NOUN
ejpam-1857	38	22	.	.	PUNCT
ejpam-1857	39	1	our	our	PRON
ejpam-1857	39	2	main	main	ADJ
ejpam-1857	39	3	motivation	motivation	NOUN
ejpam-1857	39	4	for	for	ADP
ejpam-1857	39	5	the	the	DET
ejpam-1857	39	6	study	study	NOUN
ejpam-1857	39	7	of	of	ADP
ejpam-1857	39	8	this	this	DET
ejpam-1857	39	9	family	family	NOUN
ejpam-1857	39	10	is	be	AUX
ejpam-1857	39	11	its	its	PRON
ejpam-1857	39	12	similarity	similarity	NOUN
ejpam-1857	39	13	with	with	ADP
ejpam-1857	39	14	the	the	DET
ejpam-1857	39	15	well	well	ADV
ejpam-1857	39	16	-	-	PUNCT
ejpam-1857	39	17	known	know	VERB
ejpam-1857	39	18	frey	frey	NOUN
ejpam-1857	39	19	curves	curve	NOUN
ejpam-1857	39	20	of	of	ADP
ejpam-1857	39	21	the	the	DET
ejpam-1857	39	22	form	form	NOUN
ejpam-1857	39	23	y2	y2	NOUN
ejpam-1857	39	24	=	=	SYM
ejpam-1857	39	25	x(x	x(x	PROPN
ejpam-1857	40	1	−	−	PROPN
ejpam-1857	40	2	a2)(x	a2)(x	PROPN
ejpam-1857	40	3	+	+	NUM
ejpam-1857	40	4	b2	b2	NOUN
ejpam-1857	40	5	)	)	PUNCT
ejpam-1857	40	6	(	(	PUNCT
ejpam-1857	40	7	3	3	X
ejpam-1857	40	8	)	)	PUNCT
ejpam-1857	40	9	with	with	ADP
ejpam-1857	40	10	a2	a2	PROPN
ejpam-1857	40	11	+	+	CCONJ
ejpam-1857	40	12	b2	b2	NOUN
ejpam-1857	40	13	=	=	PROPN
ejpam-1857	40	14	c2	c2	PROPN
ejpam-1857	40	15	.	.	PUNCT
ejpam-1857	41	1	2	2	X
ejpam-1857	41	2	.	.	X
ejpam-1857	41	3	results	result	NOUN
ejpam-1857	41	4	about	about	ADP
ejpam-1857	41	5	the	the	DET
ejpam-1857	41	6	new	new	ADJ
ejpam-1857	41	7	family	family	NOUN
ejpam-1857	41	8	of	of	ADP
ejpam-1857	41	9	curves	curve	NOUN
ejpam-1857	41	10	a	a	DET
ejpam-1857	41	11	primitive	primitive	ADJ
ejpam-1857	41	12	pythagorean	pythagorean	NOUN
ejpam-1857	41	13	triple	triple	NOUN
ejpam-1857	41	14	is	be	AUX
ejpam-1857	41	15	a	a	DET
ejpam-1857	41	16	triple	triple	NOUN
ejpam-1857	41	17	of	of	ADP
ejpam-1857	41	18	nonzero	nonzero	NOUN
ejpam-1857	41	19	integers	integer	NOUN
ejpam-1857	41	20	(	(	PUNCT
ejpam-1857	41	21	a	a	DET
ejpam-1857	41	22	,	,	PUNCT
ejpam-1857	41	23	b	b	NOUN
ejpam-1857	41	24	,	,	PUNCT
ejpam-1857	41	25	c	c	NOUN
ejpam-1857	41	26	)	)	PUNCT
ejpam-1857	41	27	so	so	SCONJ
ejpam-1857	41	28	that	that	SCONJ
ejpam-1857	41	29	a	a	DET
ejpam-1857	41	30	,	,	PUNCT
ejpam-1857	41	31	b	b	NOUN
ejpam-1857	41	32	,	,	PUNCT
ejpam-1857	41	33	and	and	CCONJ
ejpam-1857	41	34	c	c	NOUN
ejpam-1857	41	35	have	have	VERB
ejpam-1857	41	36	no	no	DET
ejpam-1857	41	37	common	common	ADJ
ejpam-1857	41	38	divisors	divisor	NOUN
ejpam-1857	41	39	and	and	CCONJ
ejpam-1857	41	40	satisfy	satisfy	VERB
ejpam-1857	41	41	the	the	DET
ejpam-1857	41	42	relation	relation	NOUN
ejpam-1857	41	43	a2	a2	PROPN
ejpam-1857	41	44	+	+	CCONJ
ejpam-1857	41	45	b2	b2	NOUN
ejpam-1857	41	46	=	=	PROPN
ejpam-1857	41	47	c2	c2	PROPN
ejpam-1857	41	48	.	.	PUNCT
ejpam-1857	42	1	in	in	ADP
ejpam-1857	42	2	general	general	ADJ
ejpam-1857	42	3	,	,	PUNCT
ejpam-1857	42	4	we	we	PRON
ejpam-1857	42	5	can	can	AUX
ejpam-1857	42	6	generate	generate	VERB
ejpam-1857	42	7	(	(	PUNCT
ejpam-1857	42	8	a	a	DET
ejpam-1857	42	9	,	,	PUNCT
ejpam-1857	42	10	b	b	NOUN
ejpam-1857	42	11	,	,	PUNCT
ejpam-1857	42	12	c	c	NOUN
ejpam-1857	42	13	)	)	PUNCT
ejpam-1857	42	14	by	by	ADP
ejpam-1857	42	15	the	the	DET
ejpam-1857	42	16	following	follow	VERB
ejpam-1857	42	17	relations	relation	NOUN
ejpam-1857	42	18	:	:	PUNCT
ejpam-1857	42	19	a	a	DET
ejpam-1857	42	20	=	=	PROPN
ejpam-1857	42	21	i2	i2	PROPN
ejpam-1857	42	22	−	−	PROPN
ejpam-1857	42	23	j2	j2	PROPN
ejpam-1857	42	24	,	,	PUNCT
ejpam-1857	42	25	b	b	X
ejpam-1857	42	26	=	=	SYM
ejpam-1857	42	27	2i	2i	PROPN
ejpam-1857	42	28	j	j	PROPN
ejpam-1857	42	29	,	,	PUNCT
ejpam-1857	42	30	c	c	PROPN
ejpam-1857	42	31	=	=	PROPN
ejpam-1857	42	32	i2	i2	PROPN
ejpam-1857	42	33	+	+	CCONJ
ejpam-1857	42	34	j2	j2	PROPN
ejpam-1857	42	35	,	,	PUNCT
ejpam-1857	42	36	(	(	PUNCT
ejpam-1857	42	37	4	4	X
ejpam-1857	42	38	)	)	PUNCT
ejpam-1857	43	1	where	where	SCONJ
ejpam-1857	43	2	gcd	gcd	PROPN
ejpam-1857	43	3	(	(	PUNCT
ejpam-1857	43	4	i	i	PROPN
ejpam-1857	43	5	,	,	PUNCT
ejpam-1857	43	6	j	j	PROPN
ejpam-1857	43	7	)	)	PUNCT
ejpam-1857	43	8	=	=	SYM
ejpam-1857	43	9	1	1	NUM
ejpam-1857	43	10	,	,	PUNCT
ejpam-1857	43	11	and	and	CCONJ
ejpam-1857	43	12	i	i	PRON
ejpam-1857	43	13	,	,	PUNCT
ejpam-1857	43	14	j	j	PROPN
ejpam-1857	43	15	have	have	VERB
ejpam-1857	43	16	opposite	opposite	ADJ
ejpam-1857	43	17	parity	parity	NOUN
ejpam-1857	43	18	.	.	PUNCT
ejpam-1857	44	1	throughout	throughout	ADP
ejpam-1857	44	2	,	,	PUNCT
ejpam-1857	44	3	we	we	PRON
ejpam-1857	44	4	focus	focus	VERB
ejpam-1857	44	5	on	on	ADP
ejpam-1857	44	6	the	the	DET
ejpam-1857	44	7	elliptic	elliptic	ADJ
ejpam-1857	44	8	curves	curve	NOUN
ejpam-1857	44	9	of	of	ADP
ejpam-1857	44	10	the	the	DET
ejpam-1857	44	11	form	form	NOUN
ejpam-1857	44	12	y2	y2	NOUN
ejpam-1857	44	13	=	=	SYM
ejpam-1857	44	14	x(x	x(x	PROPN
ejpam-1857	45	1	−	−	PROPN
ejpam-1857	45	2	a2)(x	a2)(x	PROPN
ejpam-1857	45	3	−	−	PROPN
ejpam-1857	45	4	b2	b2	NOUN
ejpam-1857	45	5	)	)	PUNCT
ejpam-1857	45	6	,	,	PUNCT
ejpam-1857	45	7	(	(	PUNCT
ejpam-1857	45	8	5	5	X
ejpam-1857	45	9	)	)	PUNCT
ejpam-1857	45	10	where	where	SCONJ
ejpam-1857	45	11	(	(	PUNCT
ejpam-1857	45	12	a	a	DET
ejpam-1857	45	13	,	,	PUNCT
ejpam-1857	45	14	b	b	NOUN
ejpam-1857	45	15	,	,	PUNCT
ejpam-1857	45	16	c	c	NOUN
ejpam-1857	45	17	)	)	PUNCT
ejpam-1857	45	18	is	be	AUX
ejpam-1857	45	19	a	a	DET
ejpam-1857	45	20	primitive	primitive	ADJ
ejpam-1857	45	21	pythagorean	pythagorean	NOUN
ejpam-1857	45	22	triple	triple	NOUN
ejpam-1857	45	23	.	.	PUNCT
ejpam-1857	46	1	f.	f.	PROPN
ejpam-1857	46	2	izadi	izadi	PROPN
ejpam-1857	46	3	,	,	PUNCT
ejpam-1857	46	4	k.	k.	PROPN
ejpam-1857	46	5	nabardi	nabardi	PROPN
ejpam-1857	46	6	/	/	SYM
ejpam-1857	46	7	eur	eur	PROPN
ejpam-1857	46	8	.	.	PUNCT
ejpam-1857	47	1	j.	j.	PROPN
ejpam-1857	47	2	pure	pure	PROPN
ejpam-1857	47	3	appl	appl	PROPN
ejpam-1857	47	4	.	.	PROPN
ejpam-1857	47	5	math	math	PROPN
ejpam-1857	47	6	,	,	PUNCT
ejpam-1857	47	7	7	7	NUM
ejpam-1857	47	8	(	(	PUNCT
ejpam-1857	47	9	2014	2014	NUM
ejpam-1857	47	10	)	)	PUNCT
ejpam-1857	47	11	,	,	PUNCT
ejpam-1857	47	12	131	131	NUM
ejpam-1857	47	13	-	-	SYM
ejpam-1857	47	14	139	139	NUM
ejpam-1857	47	15	133	133	NUM
ejpam-1857	47	16	lemma	lemma	PROPN
ejpam-1857	47	17	1	1	NUM
ejpam-1857	47	18	.	.	PUNCT
ejpam-1857	48	1	let	let	VERB
ejpam-1857	48	2	e	e	PRON
ejpam-1857	48	3	be	be	AUX
ejpam-1857	48	4	given	give	VERB
ejpam-1857	48	5	by	by	ADP
ejpam-1857	48	6	y2	y2	PROPN
ejpam-1857	48	7	=	=	SYM
ejpam-1857	49	1	x3	x3	ADJ
ejpam-1857	49	2	+	+	CCONJ
ejpam-1857	49	3	ax2	ax2	NOUN
ejpam-1857	49	4	+	+	CCONJ
ejpam-1857	49	5	bx	bx	NOUN
ejpam-1857	49	6	+	+	CCONJ
ejpam-1857	49	7	c	c	PROPN
ejpam-1857	49	8	and	and	CCONJ
ejpam-1857	49	9	p	p	NOUN
ejpam-1857	49	10	=	=	PUNCT
ejpam-1857	49	11	(	(	PUNCT
ejpam-1857	49	12	x	x	INTJ
ejpam-1857	49	13	,	,	PUNCT
ejpam-1857	49	14	y	y	PROPN
ejpam-1857	49	15	)	)	PUNCT
ejpam-1857	49	16	∈	∈	PROPN
ejpam-1857	49	17	e(q	e(q	PROPN
ejpam-1857	49	18	)	)	PUNCT
ejpam-1857	49	19	.	.	PUNCT
ejpam-1857	50	1	then	then	ADV
ejpam-1857	50	2	p	p	NOUN
ejpam-1857	50	3	has	have	VERB
ejpam-1857	50	4	order	order	NOUN
ejpam-1857	50	5	2	2	NUM
ejpam-1857	50	6	if	if	SCONJ
ejpam-1857	50	7	and	and	CCONJ
ejpam-1857	50	8	only	only	ADV
ejpam-1857	50	9	if	if	SCONJ
ejpam-1857	50	10	y	y	PROPN
ejpam-1857	50	11	=	=	NOUN
ejpam-1857	50	12	0	0	X
ejpam-1857	50	13	.	.	PUNCT
ejpam-1857	51	1	proof	proof	NOUN
ejpam-1857	51	2	.	.	PUNCT
ejpam-1857	52	1	please	please	INTJ
ejpam-1857	52	2	see	see	VERB
ejpam-1857	52	3	[	[	X
ejpam-1857	52	4	10	10	NUM
ejpam-1857	52	5	,	,	PUNCT
ejpam-1857	52	6	page	page	NOUN
ejpam-1857	52	7	77	77	NUM
ejpam-1857	52	8	]	]	PUNCT
ejpam-1857	52	9	.	.	PUNCT
ejpam-1857	53	1	lemma	lemma	PROPN
ejpam-1857	53	2	2	2	NUM
ejpam-1857	53	3	.	.	PUNCT
ejpam-1857	54	1	the	the	DET
ejpam-1857	54	2	elliptic	elliptic	ADJ
ejpam-1857	54	3	curve	curve	NOUN
ejpam-1857	54	4	defined	define	VERB
ejpam-1857	54	5	by	by	ADP
ejpam-1857	54	6	(	(	PUNCT
ejpam-1857	54	7	5	5	NUM
ejpam-1857	54	8	)	)	PUNCT
ejpam-1857	54	9	has	have	VERB
ejpam-1857	54	10	three	three	NUM
ejpam-1857	54	11	points	point	NOUN
ejpam-1857	54	12	of	of	ADP
ejpam-1857	54	13	order	order	NOUN
ejpam-1857	54	14	2	2	NUM
ejpam-1857	54	15	.	.	PUNCT
ejpam-1857	55	1	proof	proof	NOUN
ejpam-1857	55	2	.	.	PUNCT
ejpam-1857	56	1	it	it	PRON
ejpam-1857	56	2	is	be	AUX
ejpam-1857	56	3	clear	clear	ADJ
ejpam-1857	56	4	that	that	SCONJ
ejpam-1857	56	5	the	the	DET
ejpam-1857	56	6	points	point	NOUN
ejpam-1857	56	7	p1	p1	NOUN
ejpam-1857	56	8	=	=	SYM
ejpam-1857	56	9	(	(	PUNCT
ejpam-1857	56	10	0,0	0,0	NOUN
ejpam-1857	56	11	)	)	PUNCT
ejpam-1857	56	12	,	,	PUNCT
ejpam-1857	56	13	p2	p2	X
ejpam-1857	56	14	=	=	SYM
ejpam-1857	56	15	(	(	PUNCT
ejpam-1857	56	16	a2	a2	PROPN
ejpam-1857	56	17	,	,	PUNCT
ejpam-1857	56	18	0	0	NUM
ejpam-1857	56	19	)	)	PUNCT
ejpam-1857	56	20	,	,	PUNCT
ejpam-1857	56	21	p3	p3	PROPN
ejpam-1857	56	22	=	=	SYM
ejpam-1857	56	23	(	(	PUNCT
ejpam-1857	56	24	b2	b2	PROPN
ejpam-1857	56	25	,	,	PUNCT
ejpam-1857	56	26	0	0	NUM
ejpam-1857	56	27	)	)	PUNCT
ejpam-1857	56	28	are	be	AUX
ejpam-1857	56	29	of	of	ADP
ejpam-1857	56	30	order	order	NOUN
ejpam-1857	56	31	2	2	NUM
ejpam-1857	56	32	.	.	PUNCT
ejpam-1857	57	1	then	then	ADV
ejpam-1857	57	2	e(q)[2	e(q)[2	X
ejpam-1857	57	3	]	]	X
ejpam-1857	57	4	'	'	PUNCT
ejpam-1857	57	5	z/2z×z/2z	z/2z×z/2z	NOUN
ejpam-1857	57	6	.	.	PUNCT
ejpam-1857	58	1	we	we	PRON
ejpam-1857	58	2	wish	wish	VERB
ejpam-1857	58	3	to	to	PART
ejpam-1857	58	4	show	show	VERB
ejpam-1857	58	5	the	the	DET
ejpam-1857	58	6	torsion	torsion	NOUN
ejpam-1857	58	7	group	group	NOUN
ejpam-1857	58	8	of	of	ADP
ejpam-1857	58	9	(	(	PUNCT
ejpam-1857	58	10	5	5	NUM
ejpam-1857	58	11	)	)	PUNCT
ejpam-1857	58	12	is	be	AUX
ejpam-1857	58	13	z/2z×z/2z	z/2z×z/2z	PRON
ejpam-1857	58	14	.	.	PUNCT
ejpam-1857	59	1	so	so	ADV
ejpam-1857	59	2	we	we	PRON
ejpam-1857	59	3	have	have	VERB
ejpam-1857	59	4	to	to	PART
ejpam-1857	59	5	prove	prove	VERB
ejpam-1857	59	6	that	that	SCONJ
ejpam-1857	59	7	there	there	PRON
ejpam-1857	59	8	are	be	VERB
ejpam-1857	59	9	no	no	DET
ejpam-1857	59	10	points	point	NOUN
ejpam-1857	59	11	of	of	ADP
ejpam-1857	59	12	order	order	NOUN
ejpam-1857	59	13	4	4	NUM
ejpam-1857	59	14	,	,	PUNCT
ejpam-1857	59	15	6	6	NUM
ejpam-1857	59	16	,	,	PUNCT
ejpam-1857	59	17	and	and	CCONJ
ejpam-1857	59	18	8	8	NUM
ejpam-1857	59	19	.	.	PUNCT
ejpam-1857	60	1	in	in	ADP
ejpam-1857	60	2	order	order	NOUN
ejpam-1857	60	3	to	to	PART
ejpam-1857	60	4	show	show	VERB
ejpam-1857	60	5	that	that	SCONJ
ejpam-1857	60	6	the	the	DET
ejpam-1857	60	7	above	above	ADJ
ejpam-1857	60	8	family	family	NOUN
ejpam-1857	60	9	has	have	VERB
ejpam-1857	60	10	no	no	DET
ejpam-1857	60	11	point	point	NOUN
ejpam-1857	60	12	of	of	ADP
ejpam-1857	60	13	order	order	NOUN
ejpam-1857	60	14	4	4	NUM
ejpam-1857	60	15	,	,	PUNCT
ejpam-1857	60	16	we	we	PRON
ejpam-1857	60	17	need	need	VERB
ejpam-1857	60	18	the	the	DET
ejpam-1857	60	19	following	follow	VERB
ejpam-1857	60	20	theorem	theorem	VERB
ejpam-1857	60	21	.	.	PUNCT
ejpam-1857	60	22	theorem	theorem	NOUN
ejpam-1857	60	23	1	1	NUM
ejpam-1857	60	24	.	.	PUNCT
ejpam-1857	61	1	let	let	VERB
ejpam-1857	61	2	e	e	PRON
ejpam-1857	61	3	be	be	AUX
ejpam-1857	61	4	an	an	DET
ejpam-1857	61	5	elliptic	elliptic	ADJ
ejpam-1857	61	6	curve	curve	NOUN
ejpam-1857	61	7	defined	define	VERB
ejpam-1857	61	8	over	over	ADP
ejpam-1857	61	9	a	a	DET
ejpam-1857	61	10	field	field	NOUN
ejpam-1857	61	11	f	f	NOUN
ejpam-1857	61	12	by	by	ADP
ejpam-1857	61	13	the	the	DET
ejpam-1857	61	14	equation	equation	NOUN
ejpam-1857	61	15	y2	y2	NOUN
ejpam-1857	61	16	=	=	SYM
ejpam-1857	62	1	(	(	PUNCT
ejpam-1857	62	2	x	x	X
ejpam-1857	62	3	−α)(x	−α)(x	NOUN
ejpam-1857	62	4	−	−	PROPN
ejpam-1857	62	5	β)(x	β)(x	PUNCT
ejpam-1857	62	6	−	−	PROPN
ejpam-1857	62	7	γ	γ	X
ejpam-1857	62	8	)	)	PUNCT
ejpam-1857	62	9	=	=	PUNCT
ejpam-1857	62	10	x3	x3	ADJ
ejpam-1857	62	11	+	+	CCONJ
ejpam-1857	62	12	ax2	ax2	NOUN
ejpam-1857	62	13	+	+	CCONJ
ejpam-1857	62	14	bx	bx	NOUN
ejpam-1857	63	1	+	+	CCONJ
ejpam-1857	63	2	c	c	X
ejpam-1857	63	3	,	,	PUNCT
ejpam-1857	63	4	where	where	SCONJ
ejpam-1857	63	5	char(f	char(f	NOUN
ejpam-1857	63	6	)	)	PUNCT
ejpam-1857	63	7	6=	6=	ADP
ejpam-1857	63	8	2	2	X
ejpam-1857	63	9	.	.	X
ejpam-1857	64	1	for	for	ADP
ejpam-1857	64	2	(	(	PUNCT
ejpam-1857	64	3	x	x	NOUN
ejpam-1857	64	4	′	′	PROPN
ejpam-1857	64	5	,	,	PUNCT
ejpam-1857	64	6	y	y	PROPN
ejpam-1857	64	7	′	′	NOUN
ejpam-1857	64	8	)	)	PUNCT
ejpam-1857	64	9	∈	∈	PROPN
ejpam-1857	64	10	e(f	e(f	PROPN
ejpam-1857	64	11	)	)	PUNCT
ejpam-1857	64	12	,	,	PUNCT
ejpam-1857	64	13	there	there	PRON
ejpam-1857	64	14	exists	exist	VERB
ejpam-1857	64	15	(	(	PUNCT
ejpam-1857	64	16	x	x	X
ejpam-1857	64	17	,	,	PUNCT
ejpam-1857	64	18	y	y	PROPN
ejpam-1857	64	19	)	)	PUNCT
ejpam-1857	64	20	∈	∈	PROPN
ejpam-1857	64	21	e(f	e(f	PROPN
ejpam-1857	64	22	)	)	PUNCT
ejpam-1857	64	23	with	with	ADP
ejpam-1857	64	24	2(x	2(x	NUM
ejpam-1857	64	25	,	,	PUNCT
ejpam-1857	64	26	y	y	NOUN
ejpam-1857	64	27	)	)	PUNCT
ejpam-1857	64	28	=	=	SYM
ejpam-1857	65	1	(	(	PUNCT
ejpam-1857	65	2	x	x	SYM
ejpam-1857	65	3	′	′	NOUN
ejpam-1857	65	4	,	,	PUNCT
ejpam-1857	65	5	y	y	PROPN
ejpam-1857	65	6	′	′	NUM
ejpam-1857	65	7	)	)	PUNCT
ejpam-1857	65	8	,	,	PUNCT
ejpam-1857	65	9	if	if	SCONJ
ejpam-1857	65	10	and	and	CCONJ
ejpam-1857	65	11	only	only	ADV
ejpam-1857	65	12	if	if	SCONJ
ejpam-1857	65	13	x	x	NUM
ejpam-1857	65	14	′	′	NUM
ejpam-1857	65	15	−α	−α	NOUN
ejpam-1857	65	16	,	,	PUNCT
ejpam-1857	65	17	x	x	NOUN
ejpam-1857	66	1	′	′	NOUN
ejpam-1857	66	2	−	−	NUM
ejpam-1857	66	3	β	β	X
ejpam-1857	66	4	,	,	PUNCT
ejpam-1857	66	5	and	and	CCONJ
ejpam-1857	66	6	x	x	X
ejpam-1857	67	1	′	′	NOUN
ejpam-1857	67	2	−	−	NOUN
ejpam-1857	67	3	γ	γ	NOUN
ejpam-1857	67	4	are	be	AUX
ejpam-1857	67	5	squares	square	NOUN
ejpam-1857	67	6	.	.	PUNCT
ejpam-1857	68	1	proof	proof	NOUN
ejpam-1857	68	2	.	.	PUNCT
ejpam-1857	69	1	see	see	VERB
ejpam-1857	69	2	[	[	X
ejpam-1857	69	3	2	2	NUM
ejpam-1857	69	4	,	,	PUNCT
ejpam-1857	69	5	theorem	theorem	VERB
ejpam-1857	69	6	4.1	4.1	NUM
ejpam-1857	69	7	,	,	PUNCT
ejpam-1857	69	8	page	page	NOUN
ejpam-1857	69	9	37	37	NUM
ejpam-1857	69	10	]	]	PUNCT
ejpam-1857	69	11	.	.	PUNCT
ejpam-1857	70	1	applying	apply	VERB
ejpam-1857	70	2	this	this	DET
ejpam-1857	70	3	theorem	theorem	NOUN
ejpam-1857	70	4	to	to	ADP
ejpam-1857	70	5	our	our	PRON
ejpam-1857	70	6	family	family	NOUN
ejpam-1857	70	7	,	,	PUNCT
ejpam-1857	70	8	we	we	PRON
ejpam-1857	70	9	get	get	VERB
ejpam-1857	70	10	the	the	DET
ejpam-1857	70	11	following	follow	VERB
ejpam-1857	70	12	result	result	NOUN
ejpam-1857	70	13	.	.	PUNCT
ejpam-1857	71	1	proposition	proposition	NOUN
ejpam-1857	71	2	1	1	NUM
ejpam-1857	71	3	.	.	PUNCT
ejpam-1857	72	1	the	the	DET
ejpam-1857	72	2	elliptic	elliptic	ADJ
ejpam-1857	72	3	curve	curve	NOUN
ejpam-1857	72	4	in	in	ADP
ejpam-1857	72	5	the	the	DET
ejpam-1857	72	6	form	form	NOUN
ejpam-1857	72	7	(	(	PUNCT
ejpam-1857	72	8	5	5	NUM
ejpam-1857	72	9	)	)	PUNCT
ejpam-1857	72	10	does	do	AUX
ejpam-1857	72	11	not	not	PART
ejpam-1857	72	12	have	have	VERB
ejpam-1857	72	13	any	any	DET
ejpam-1857	72	14	points	point	NOUN
ejpam-1857	72	15	of	of	ADP
ejpam-1857	72	16	order	order	NOUN
ejpam-1857	72	17	4	4	NUM
ejpam-1857	72	18	.	.	PUNCT
ejpam-1857	73	1	proof	proof	NOUN
ejpam-1857	73	2	.	.	PUNCT
ejpam-1857	74	1	let	let	VERB
ejpam-1857	74	2	p	p	NOUN
ejpam-1857	74	3	=	=	X
ejpam-1857	74	4	(	(	PUNCT
ejpam-1857	74	5	x	x	INTJ
ejpam-1857	74	6	,	,	PUNCT
ejpam-1857	74	7	y	y	PROPN
ejpam-1857	74	8	)	)	PUNCT
ejpam-1857	74	9	∈	∈	PROPN
ejpam-1857	74	10	e(q	e(q	PROPN
ejpam-1857	74	11	)	)	PUNCT
ejpam-1857	74	12	be	be	VERB
ejpam-1857	74	13	such	such	ADJ
ejpam-1857	74	14	that	that	PRON
ejpam-1857	74	15	4p	4p	NUM
ejpam-1857	75	1	=	=	SYM
ejpam-1857	75	2	o	o	NOUN
ejpam-1857	75	3	.	.	PUNCT
ejpam-1857	76	1	then	then	ADV
ejpam-1857	76	2	one	one	NUM
ejpam-1857	76	3	of	of	ADP
ejpam-1857	76	4	following	follow	VERB
ejpam-1857	76	5	cases	case	NOUN
ejpam-1857	76	6	must	must	AUX
ejpam-1857	76	7	be	be	AUX
ejpam-1857	76	8	true	true	ADJ
ejpam-1857	76	9	:	:	PUNCT
ejpam-1857	76	10	2p	2p	NUM
ejpam-1857	76	11	=	=	SYM
ejpam-1857	76	12	(	(	PUNCT
ejpam-1857	76	13	0,0	0,0	NOUN
ejpam-1857	76	14	)	)	PUNCT
ejpam-1857	76	15	,	,	PUNCT
ejpam-1857	76	16	2p	2p	NUM
ejpam-1857	76	17	=	=	SYM
ejpam-1857	76	18	(	(	PUNCT
ejpam-1857	76	19	a2	a2	PROPN
ejpam-1857	76	20	,	,	PUNCT
ejpam-1857	76	21	0	0	NUM
ejpam-1857	76	22	)	)	PUNCT
ejpam-1857	76	23	,	,	PUNCT
ejpam-1857	76	24	2p	2p	NUM
ejpam-1857	76	25	=	=	SYM
ejpam-1857	76	26	(	(	PUNCT
ejpam-1857	76	27	b2	b2	NOUN
ejpam-1857	76	28	,	,	PUNCT
ejpam-1857	76	29	0	0	NUM
ejpam-1857	76	30	)	)	PUNCT
ejpam-1857	76	31	.	.	PUNCT
ejpam-1857	77	1	if	if	SCONJ
ejpam-1857	77	2	2p	2p	NUM
ejpam-1857	77	3	=	=	SYM
ejpam-1857	77	4	(	(	PUNCT
ejpam-1857	77	5	0	0	NUM
ejpam-1857	77	6	,	,	PUNCT
ejpam-1857	77	7	0	0	NUM
ejpam-1857	77	8	)	)	PUNCT
ejpam-1857	77	9	,	,	PUNCT
ejpam-1857	77	10	then	then	ADV
ejpam-1857	77	11	−a2	−a2	PROPN
ejpam-1857	77	12	and	and	CCONJ
ejpam-1857	77	13	−b2	−b2	PROPN
ejpam-1857	77	14	are	be	AUX
ejpam-1857	77	15	squares	square	NOUN
ejpam-1857	77	16	which	which	PRON
ejpam-1857	77	17	are	be	AUX
ejpam-1857	77	18	contradiction	contradiction	NOUN
ejpam-1857	77	19	.	.	PUNCT
ejpam-1857	78	1	let	let	VERB
ejpam-1857	78	2	2p	2p	NUM
ejpam-1857	78	3	=	=	SYM
ejpam-1857	78	4	(	(	PUNCT
ejpam-1857	78	5	a2	a2	PROPN
ejpam-1857	78	6	,	,	PUNCT
ejpam-1857	78	7	0	0	NUM
ejpam-1857	78	8	)	)	PUNCT
ejpam-1857	78	9	,	,	PUNCT
ejpam-1857	78	10	then	then	ADV
ejpam-1857	78	11	a2	a2	PROPN
ejpam-1857	78	12	−	−	PROPN
ejpam-1857	78	13	b2	b2	NOUN
ejpam-1857	78	14	is	be	AUX
ejpam-1857	78	15	a	a	DET
ejpam-1857	78	16	square	square	NOUN
ejpam-1857	78	17	.	.	PUNCT
ejpam-1857	79	1	so	so	ADV
ejpam-1857	79	2	we	we	PRON
ejpam-1857	79	3	have	have	VERB
ejpam-1857	79	4	,	,	PUNCT
ejpam-1857	80	1	a2	a2	PROPN
ejpam-1857	80	2	−	−	PROPN
ejpam-1857	80	3	b2	b2	NOUN
ejpam-1857	80	4	=	=	SYM
ejpam-1857	80	5	d2	d2	PROPN
ejpam-1857	80	6	for	for	ADP
ejpam-1857	80	7	some	some	DET
ejpam-1857	80	8	d	d	PROPN
ejpam-1857	80	9	∈	∈	PROPN
ejpam-1857	80	10	z	z	PROPN
ejpam-1857	80	11	and	and	CCONJ
ejpam-1857	80	12	a2	a2	PROPN
ejpam-1857	80	13	+	+	CCONJ
ejpam-1857	80	14	b2	b2	NOUN
ejpam-1857	80	15	=	=	PROPN
ejpam-1857	80	16	c2	c2	PROPN
ejpam-1857	80	17	.	.	PUNCT
ejpam-1857	81	1	therefore	therefore	ADV
ejpam-1857	81	2	(	(	PUNCT
ejpam-1857	81	3	a	a	DET
ejpam-1857	81	4	b	b	NOUN
ejpam-1857	81	5	)	)	PUNCT
ejpam-1857	81	6	2−1=	2−1=	NUM
ejpam-1857	82	1	(	(	PUNCT
ejpam-1857	82	2	d	d	NOUN
ejpam-1857	82	3	b	b	PROPN
ejpam-1857	82	4	)	)	PUNCT
ejpam-1857	82	5	2	2	NUM
ejpam-1857	82	6	and	and	CCONJ
ejpam-1857	82	7	(	(	PUNCT
ejpam-1857	82	8	a	a	DET
ejpam-1857	82	9	b	b	NOUN
ejpam-1857	82	10	)	)	PUNCT
ejpam-1857	82	11	2	2	NUM
ejpam-1857	83	1	+	+	NOUN
ejpam-1857	83	2	1=	1=	NUM
ejpam-1857	83	3	(	(	PUNCT
ejpam-1857	83	4	c	c	PROPN
ejpam-1857	83	5	b	b	PROPN
ejpam-1857	83	6	)	)	PUNCT
ejpam-1857	83	7	2	2	NUM
ejpam-1857	83	8	.	.	PUNCT
ejpam-1857	84	1	this	this	PRON
ejpam-1857	84	2	means	mean	VERB
ejpam-1857	84	3	that	that	SCONJ
ejpam-1857	84	4	1	1	NUM
ejpam-1857	84	5	is	be	AUX
ejpam-1857	84	6	a	a	DET
ejpam-1857	84	7	congruent	congruent	ADJ
ejpam-1857	84	8	number	number	NOUN
ejpam-1857	84	9	again	again	ADV
ejpam-1857	84	10	a	a	DET
ejpam-1857	84	11	contradiction	contradiction	NOUN
ejpam-1857	84	12	.	.	PUNCT
ejpam-1857	85	1	the	the	DET
ejpam-1857	85	2	case	case	NOUN
ejpam-1857	85	3	2p	2p	NUM
ejpam-1857	85	4	=	=	SYM
ejpam-1857	85	5	(	(	PUNCT
ejpam-1857	85	6	b2	b2	PROPN
ejpam-1857	85	7	,	,	PUNCT
ejpam-1857	85	8	0	0	NUM
ejpam-1857	85	9	)	)	PUNCT
ejpam-1857	85	10	is	be	AUX
ejpam-1857	85	11	similar	similar	ADJ
ejpam-1857	85	12	.	.	PUNCT
ejpam-1857	86	1	corollary	corollary	ADJ
ejpam-1857	86	2	1	1	NUM
ejpam-1857	86	3	.	.	PUNCT
ejpam-1857	87	1	there	there	PRON
ejpam-1857	87	2	is	be	VERB
ejpam-1857	87	3	no	no	DET
ejpam-1857	87	4	points	point	NOUN
ejpam-1857	87	5	of	of	ADP
ejpam-1857	87	6	order	order	NOUN
ejpam-1857	87	7	8	8	NUM
ejpam-1857	87	8	on	on	ADP
ejpam-1857	87	9	(	(	PUNCT
ejpam-1857	87	10	5	5	NUM
ejpam-1857	87	11	)	)	PUNCT
ejpam-1857	87	12	.	.	PUNCT
ejpam-1857	88	1	the	the	DET
ejpam-1857	88	2	following	follow	VERB
ejpam-1857	88	3	proposition	proposition	NOUN
ejpam-1857	88	4	is	be	AUX
ejpam-1857	88	5	also	also	ADV
ejpam-1857	88	6	necessary	necessary	ADJ
ejpam-1857	88	7	.	.	PUNCT
ejpam-1857	89	1	proposition	proposition	NOUN
ejpam-1857	89	2	2	2	NUM
ejpam-1857	89	3	.	.	PUNCT
ejpam-1857	90	1	the	the	DET
ejpam-1857	90	2	elliptic	elliptic	ADJ
ejpam-1857	90	3	curve	curve	NOUN
ejpam-1857	90	4	in	in	ADP
ejpam-1857	90	5	the	the	DET
ejpam-1857	90	6	form	form	NOUN
ejpam-1857	90	7	(	(	PUNCT
ejpam-1857	90	8	5	5	NUM
ejpam-1857	90	9	)	)	PUNCT
ejpam-1857	90	10	does	do	AUX
ejpam-1857	90	11	not	not	PART
ejpam-1857	90	12	have	have	VERB
ejpam-1857	90	13	any	any	DET
ejpam-1857	90	14	points	point	NOUN
ejpam-1857	90	15	of	of	ADP
ejpam-1857	90	16	order	order	NOUN
ejpam-1857	90	17	6	6	NUM
ejpam-1857	90	18	.	.	PUNCT
ejpam-1857	91	1	proof	proof	NOUN
ejpam-1857	91	2	.	.	PUNCT
ejpam-1857	92	1	by	by	ADP
ejpam-1857	92	2	interchanging	interchange	VERB
ejpam-1857	92	3	a	a	PRON
ejpam-1857	92	4	and	and	CCONJ
ejpam-1857	92	5	b	b	NOUN
ejpam-1857	92	6	if	if	SCONJ
ejpam-1857	92	7	necessary	necessary	ADJ
ejpam-1857	92	8	,	,	PUNCT
ejpam-1857	92	9	one	one	PRON
ejpam-1857	92	10	may	may	AUX
ejpam-1857	92	11	assume	assume	VERB
ejpam-1857	92	12	a2	a2	PROPN
ejpam-1857	92	13	<	<	X
ejpam-1857	92	14	b2	b2	PROPN
ejpam-1857	92	15	.	.	PUNCT
ejpam-1857	93	1	then	then	ADV
ejpam-1857	93	2	after	after	SCONJ
ejpam-1857	93	3	one	one	NUM
ejpam-1857	93	4	replaces	replace	VERB
ejpam-1857	93	5	x	x	PUNCT
ejpam-1857	93	6	by	by	ADP
ejpam-1857	93	7	x	x	SYM
ejpam-1857	93	8	+	+	NUM
ejpam-1857	93	9	b2	b2	NOUN
ejpam-1857	93	10	,	,	PUNCT
ejpam-1857	93	11	the	the	DET
ejpam-1857	93	12	elliptic	elliptic	ADJ
ejpam-1857	93	13	curve	curve	NOUN
ejpam-1857	93	14	e	e	NOUN
ejpam-1857	93	15	is	be	AUX
ejpam-1857	93	16	given	give	VERB
ejpam-1857	93	17	by	by	ADP
ejpam-1857	93	18	y2	y2	PROPN
ejpam-1857	93	19	=	=	SYM
ejpam-1857	93	20	x(x	x(x	PROPN
ejpam-1857	94	1	+	+	CCONJ
ejpam-1857	94	2	b2)(x	b2)(x	NOUN
ejpam-1857	94	3	+	+	CCONJ
ejpam-1857	94	4	b2	b2	NOUN
ejpam-1857	94	5	−	−	PROPN
ejpam-1857	94	6	a2	a2	PROPN
ejpam-1857	94	7	)	)	PUNCT
ejpam-1857	94	8	.	.	PUNCT
ejpam-1857	95	1	f.	f.	PROPN
ejpam-1857	95	2	izadi	izadi	PROPN
ejpam-1857	95	3	,	,	PUNCT
ejpam-1857	95	4	k.	k.	PROPN
ejpam-1857	95	5	nabardi	nabardi	PROPN
ejpam-1857	95	6	/	/	SYM
ejpam-1857	95	7	eur	eur	PROPN
ejpam-1857	95	8	.	.	PUNCT
ejpam-1857	96	1	j.	j.	PROPN
ejpam-1857	96	2	pure	pure	PROPN
ejpam-1857	96	3	appl	appl	PROPN
ejpam-1857	96	4	.	.	PROPN
ejpam-1857	96	5	math	math	PROPN
ejpam-1857	96	6	,	,	PUNCT
ejpam-1857	96	7	7	7	NUM
ejpam-1857	96	8	(	(	PUNCT
ejpam-1857	96	9	2014	2014	NUM
ejpam-1857	96	10	)	)	PUNCT
ejpam-1857	96	11	,	,	PUNCT
ejpam-1857	96	12	131	131	NUM
ejpam-1857	96	13	-	-	SYM
ejpam-1857	96	14	139	139	NUM
ejpam-1857	96	15	134	134	NUM
ejpam-1857	96	16	applying	apply	VERB
ejpam-1857	96	17	a	a	DET
ejpam-1857	96	18	result	result	NOUN
ejpam-1857	96	19	of	of	ADP
ejpam-1857	96	20	ono	ono	PROPN
ejpam-1857	96	21	[	[	X
ejpam-1857	96	22	5	5	NUM
ejpam-1857	96	23	,	,	PUNCT
ejpam-1857	96	24	main	main	ADJ
ejpam-1857	96	25	theorem	theorem	NOUN
ejpam-1857	96	26	1	1	NUM
ejpam-1857	96	27	]	]	PUNCT
ejpam-1857	96	28	to	to	ADP
ejpam-1857	96	29	this	this	DET
ejpam-1857	96	30	equation	equation	NOUN
ejpam-1857	96	31	,	,	PUNCT
ejpam-1857	96	32	it	it	PRON
ejpam-1857	96	33	follows	follow	VERB
ejpam-1857	96	34	that	that	SCONJ
ejpam-1857	96	35	e(q	e(q	NOUN
ejpam-1857	96	36	)	)	PUNCT
ejpam-1857	96	37	contains	contain	VERB
ejpam-1857	96	38	a	a	DET
ejpam-1857	96	39	point	point	NOUN
ejpam-1857	96	40	of	of	ADP
ejpam-1857	96	41	order	order	NOUN
ejpam-1857	96	42	6	6	NUM
ejpam-1857	96	43	iff	iff	PROPN
ejpam-1857	96	44	a	a	PRON
ejpam-1857	96	45	,	,	PUNCT
ejpam-1857	96	46	b	b	PROPN
ejpam-1857	96	47	∈	∈	PROPN
ejpam-1857	96	48	z	z	NOUN
ejpam-1857	96	49	exist	exist	VERB
ejpam-1857	96	50	satisfying	satisfy	VERB
ejpam-1857	96	51	¨	¨	ADJ
ejpam-1857	96	52	a4	a4	NOUN
ejpam-1857	96	53	+	+	CCONJ
ejpam-1857	96	54	2a3b	2a3b	ADJ
ejpam-1857	96	55	=	=	SYM
ejpam-1857	96	56	b2	b2	NOUN
ejpam-1857	96	57	;	;	PUNCT
ejpam-1857	96	58	b4	b4	NOUN
ejpam-1857	96	59	+	+	CCONJ
ejpam-1857	96	60	2b3a=	2b3a=	NUM
ejpam-1857	96	61	b2	b2	NOUN
ejpam-1857	96	62	−	−	PROPN
ejpam-1857	96	63	a2	a2	PROPN
ejpam-1857	96	64	.	.	PUNCT
ejpam-1857	97	1	recall	recall	VERB
ejpam-1857	97	2	that	that	PRON
ejpam-1857	97	3	without	without	ADP
ejpam-1857	97	4	loss	loss	NOUN
ejpam-1857	97	5	of	of	ADP
ejpam-1857	97	6	generality	generality	NOUN
ejpam-1857	97	7	we	we	PRON
ejpam-1857	97	8	can	can	AUX
ejpam-1857	97	9	assume	assume	VERB
ejpam-1857	97	10	(	(	PUNCT
ejpam-1857	97	11	a	a	DET
ejpam-1857	97	12	,	,	PUNCT
ejpam-1857	97	13	b	b	NOUN
ejpam-1857	97	14	,	,	PUNCT
ejpam-1857	97	15	c	c	NOUN
ejpam-1857	97	16	)	)	PUNCT
ejpam-1857	97	17	to	to	PART
ejpam-1857	97	18	be	be	AUX
ejpam-1857	97	19	a	a	DET
ejpam-1857	97	20	primitive	primitive	ADJ
ejpam-1857	97	21	pythagorean	pythagorean	NOUN
ejpam-1857	97	22	triple	triple	NOUN
ejpam-1857	97	23	.	.	PUNCT
ejpam-1857	98	1	then	then	ADV
ejpam-1857	98	2	only	only	ADV
ejpam-1857	98	3	two	two	NUM
ejpam-1857	98	4	possibilities	possibility	NOUN
ejpam-1857	98	5	may	may	AUX
ejpam-1857	98	6	occur	occur	VERB
ejpam-1857	98	7	:	:	PUNCT
ejpam-1857	98	8	if	if	SCONJ
ejpam-1857	98	9	b	b	NOUN
ejpam-1857	98	10	is	be	AUX
ejpam-1857	98	11	even	even	ADV
ejpam-1857	98	12	,	,	PUNCT
ejpam-1857	98	13	then	then	ADV
ejpam-1857	98	14	a	a	PRON
ejpam-1857	98	15	is	be	AUX
ejpam-1857	98	16	odd	odd	ADJ
ejpam-1857	98	17	,	,	PUNCT
ejpam-1857	98	18	and	and	CCONJ
ejpam-1857	98	19	modulo	modulo	VERB
ejpam-1857	98	20	4	4	NUM
ejpam-1857	98	21	the	the	DET
ejpam-1857	98	22	above	above	ADJ
ejpam-1857	98	23	system	system	NOUN
ejpam-1857	98	24	looks	look	VERB
ejpam-1857	98	25	like	like	ADP
ejpam-1857	98	26	¨	¨	NOUN
ejpam-1857	98	27	a4	a4	NOUN
ejpam-1857	98	28	+	+	CCONJ
ejpam-1857	98	29	2a3b	2a3b	ADJ
ejpam-1857	98	30	≡	≡	PROPN
ejpam-1857	98	31	0	0	PUNCT
ejpam-1857	99	1	(	(	PUNCT
ejpam-1857	99	2	mod	mod	PROPN
ejpam-1857	99	3	4	4	NUM
ejpam-1857	99	4	)	)	PUNCT
ejpam-1857	99	5	;	;	PUNCT
ejpam-1857	99	6	b4	b4	NOUN
ejpam-1857	99	7	+	+	CCONJ
ejpam-1857	99	8	2b3a≡	2b3a≡	NUM
ejpam-1857	99	9	−1	−1	NOUN
ejpam-1857	99	10	(	(	PUNCT
ejpam-1857	99	11	mod	mod	PROPN
ejpam-1857	99	12	4	4	NUM
ejpam-1857	99	13	)	)	PUNCT
ejpam-1857	99	14	.	.	PUNCT
ejpam-1857	100	1	this	this	PRON
ejpam-1857	100	2	is	be	AUX
ejpam-1857	100	3	impossible	impossible	ADJ
ejpam-1857	100	4	since	since	SCONJ
ejpam-1857	100	5	it	it	PRON
ejpam-1857	100	6	implies	imply	VERB
ejpam-1857	100	7	that	that	SCONJ
ejpam-1857	100	8	a	a	PRON
ejpam-1857	100	9	is	be	AUX
ejpam-1857	100	10	even	even	ADV
ejpam-1857	100	11	and	and	CCONJ
ejpam-1857	100	12	hence	hence	ADV
ejpam-1857	100	13	−1	−1	NOUN
ejpam-1857	100	14	would	would	AUX
ejpam-1857	100	15	be	be	AUX
ejpam-1857	100	16	a	a	DET
ejpam-1857	100	17	square	square	ADJ
ejpam-1857	100	18	modulo	modulo	NOUN
ejpam-1857	100	19	4	4	NUM
ejpam-1857	100	20	.	.	PUNCT
ejpam-1857	100	21	similarly	similarly	ADV
ejpam-1857	100	22	,	,	PUNCT
ejpam-1857	100	23	if	if	SCONJ
ejpam-1857	100	24	b	b	PROPN
ejpam-1857	100	25	is	be	AUX
ejpam-1857	100	26	odd	odd	ADJ
ejpam-1857	100	27	,	,	PUNCT
ejpam-1857	100	28	then	then	ADV
ejpam-1857	100	29	a	a	PRON
ejpam-1857	100	30	is	be	AUX
ejpam-1857	100	31	even	even	ADV
ejpam-1857	100	32	since	since	SCONJ
ejpam-1857	100	33	otherwise	otherwise	ADV
ejpam-1857	100	34	a2	a2	PROPN
ejpam-1857	100	35	+	+	CCONJ
ejpam-1857	100	36	b2	b2	NOUN
ejpam-1857	100	37	can	can	AUX
ejpam-1857	100	38	not	not	PART
ejpam-1857	100	39	be	be	AUX
ejpam-1857	100	40	a	a	DET
ejpam-1857	100	41	square	square	NOUN
ejpam-1857	100	42	.	.	PUNCT
ejpam-1857	101	1	hence	hence	ADV
ejpam-1857	101	2	one	one	NUM
ejpam-1857	101	3	obtains	obtain	VERB
ejpam-1857	101	4	the	the	DET
ejpam-1857	101	5	system	system	NOUN
ejpam-1857	101	6	¨	¨	NOUN
ejpam-1857	101	7	a4	a4	NOUN
ejpam-1857	101	8	+	+	CCONJ
ejpam-1857	101	9	2a3b	2a3b	ADJ
ejpam-1857	101	10	≡	≡	PROPN
ejpam-1857	101	11	1	1	NUM
ejpam-1857	101	12	(	(	PUNCT
ejpam-1857	101	13	mod	mod	NOUN
ejpam-1857	101	14	4	4	NUM
ejpam-1857	101	15	)	)	PUNCT
ejpam-1857	101	16	;	;	PUNCT
ejpam-1857	101	17	b4	b4	NOUN
ejpam-1857	101	18	+	+	CCONJ
ejpam-1857	101	19	2b3a≡	2b3a≡	NUM
ejpam-1857	101	20	1	1	NUM
ejpam-1857	101	21	(	(	PUNCT
ejpam-1857	101	22	mod	mod	NOUN
ejpam-1857	101	23	4	4	NUM
ejpam-1857	101	24	)	)	PUNCT
ejpam-1857	101	25	implying	imply	VERB
ejpam-1857	101	26	that	that	SCONJ
ejpam-1857	101	27	both	both	CCONJ
ejpam-1857	101	28	a	a	PRON
ejpam-1857	101	29	and	and	CCONJ
ejpam-1857	101	30	b	b	NOUN
ejpam-1857	101	31	are	be	AUX
ejpam-1857	101	32	odd	odd	ADJ
ejpam-1857	101	33	.	.	PUNCT
ejpam-1857	102	1	so	so	ADV
ejpam-1857	102	2	it	it	PRON
ejpam-1857	102	3	would	would	AUX
ejpam-1857	102	4	follow	follow	VERB
ejpam-1857	102	5	that	that	SCONJ
ejpam-1857	102	6	1	1	NUM
ejpam-1857	102	7	+	+	NOUN
ejpam-1857	102	8	2≡	2≡	NUM
ejpam-1857	102	9	1	1	NUM
ejpam-1857	102	10	(	(	PUNCT
ejpam-1857	102	11	mod	mod	NOUN
ejpam-1857	102	12	4	4	NUM
ejpam-1857	102	13	)	)	PUNCT
ejpam-1857	102	14	which	which	PRON
ejpam-1857	102	15	is	be	AUX
ejpam-1857	102	16	again	again	ADV
ejpam-1857	102	17	a	a	DET
ejpam-1857	102	18	contradiction	contradiction	NOUN
ejpam-1857	102	19	.	.	PUNCT
ejpam-1857	103	1	now	now	ADV
ejpam-1857	103	2	the	the	DET
ejpam-1857	103	3	following	follow	VERB
ejpam-1857	103	4	corollary	corollary	NOUN
ejpam-1857	103	5	is	be	AUX
ejpam-1857	103	6	immediate	immediate	ADJ
ejpam-1857	103	7	.	.	PUNCT
ejpam-1857	104	1	corollary	corollary	ADJ
ejpam-1857	104	2	2	2	NUM
ejpam-1857	104	3	.	.	PUNCT
ejpam-1857	104	4	e(q)tors	e(q)tor	NOUN
ejpam-1857	104	5	is	be	AUX
ejpam-1857	104	6	isomorphic	isomorphic	ADJ
ejpam-1857	104	7	to	to	ADP
ejpam-1857	104	8	z/2z×z/2z	z/2z×z/2z	PROPN
ejpam-1857	104	9	.	.	PUNCT
ejpam-1857	105	1	theorem	theorem	NOUN
ejpam-1857	105	2	2	2	NUM
ejpam-1857	105	3	.	.	X
ejpam-1857	105	4	for	for	ADP
ejpam-1857	105	5	each	each	DET
ejpam-1857	105	6	pythagorean	pythagorean	PROPN
ejpam-1857	105	7	triple	triple	NOUN
ejpam-1857	105	8	(	(	PUNCT
ejpam-1857	105	9	a	a	DET
ejpam-1857	105	10	,	,	PUNCT
ejpam-1857	105	11	b	b	NOUN
ejpam-1857	105	12	,	,	PUNCT
ejpam-1857	105	13	c	c	NOUN
ejpam-1857	105	14	)	)	PUNCT
ejpam-1857	105	15	,	,	PUNCT
ejpam-1857	105	16	the	the	DET
ejpam-1857	105	17	elliptic	elliptic	ADJ
ejpam-1857	105	18	curve	curve	NOUN
ejpam-1857	105	19	y2	y2	PROPN
ejpam-1857	105	20	=	=	SYM
ejpam-1857	105	21	x(x	x(x	PROPN
ejpam-1857	106	1	−	−	PROPN
ejpam-1857	106	2	a2)(x	a2)(x	PROPN
ejpam-1857	106	3	−	−	PROPN
ejpam-1857	106	4	b2	b2	NOUN
ejpam-1857	106	5	)	)	PUNCT
ejpam-1857	106	6	has	have	VERB
ejpam-1857	106	7	a	a	DET
ejpam-1857	106	8	positive	positive	ADJ
ejpam-1857	106	9	rank	rank	NOUN
ejpam-1857	106	10	.	.	PUNCT
ejpam-1857	107	1	proof	proof	NOUN
ejpam-1857	107	2	.	.	PUNCT
ejpam-1857	108	1	it	it	PRON
ejpam-1857	108	2	suffices	suffice	VERB
ejpam-1857	108	3	to	to	PART
ejpam-1857	108	4	consider	consider	VERB
ejpam-1857	108	5	the	the	DET
ejpam-1857	108	6	point	point	NOUN
ejpam-1857	108	7	q	q	X
ejpam-1857	109	1	=	=	SYM
ejpam-1857	109	2	(	(	PUNCT
ejpam-1857	109	3	c2	c2	PROPN
ejpam-1857	109	4	,	,	PUNCT
ejpam-1857	109	5	abc	abc	PROPN
ejpam-1857	109	6	)	)	PUNCT
ejpam-1857	109	7	on	on	ADP
ejpam-1857	109	8	(	(	PUNCT
ejpam-1857	109	9	5	5	NUM
ejpam-1857	109	10	)	)	PUNCT
ejpam-1857	109	11	.	.	PUNCT
ejpam-1857	110	1	clearly	clearly	ADV
ejpam-1857	110	2	q	q	PROPN
ejpam-1857	110	3	6∈	6∈	PROPN
ejpam-1857	110	4	{	{	PUNCT
ejpam-1857	110	5	p1	p1	PROPN
ejpam-1857	110	6	,	,	PUNCT
ejpam-1857	110	7	p2	p2	NOUN
ejpam-1857	110	8	,	,	PUNCT
ejpam-1857	110	9	p3,o	p3,o	PROPN
ejpam-1857	110	10	}	}	PUNCT
ejpam-1857	110	11	and	and	CCONJ
ejpam-1857	110	12	so	so	ADV
ejpam-1857	110	13	it	it	PRON
ejpam-1857	110	14	is	be	AUX
ejpam-1857	110	15	a	a	DET
ejpam-1857	110	16	point	point	NOUN
ejpam-1857	110	17	of	of	ADP
ejpam-1857	110	18	infinite	infinite	ADJ
ejpam-1857	110	19	order	order	NOUN
ejpam-1857	110	20	.	.	PUNCT
ejpam-1857	111	1	after	after	ADP
ejpam-1857	111	2	searching	search	VERB
ejpam-1857	111	3	through	through	ADP
ejpam-1857	111	4	202,461	202,461	NUM
ejpam-1857	111	5	curves	curve	NOUN
ejpam-1857	111	6	with	with	ADP
ejpam-1857	111	7	i	i	PRON
ejpam-1857	111	8	,	,	PUNCT
ejpam-1857	111	9	j	j	PROPN
ejpam-1857	111	10	≤	≤	PROPN
ejpam-1857	111	11	1000	1000	NUM
ejpam-1857	111	12	,	,	PUNCT
ejpam-1857	111	13	we	we	PRON
ejpam-1857	111	14	found	find	VERB
ejpam-1857	111	15	53	53	NUM
ejpam-1857	111	16	curves	curve	NOUN
ejpam-1857	111	17	of	of	ADP
ejpam-1857	111	18	rank	rank	NOUN
ejpam-1857	111	19	5	5	NUM
ejpam-1857	111	20	.	.	PUNCT
ejpam-1857	112	1	the	the	DET
ejpam-1857	112	2	first	first	ADJ
ejpam-1857	112	3	curve	curve	NOUN
ejpam-1857	112	4	with	with	ADP
ejpam-1857	112	5	rank	rank	NOUN
ejpam-1857	112	6	5	5	NUM
ejpam-1857	112	7	is	be	AUX
ejpam-1857	112	8	generated	generate	VERB
ejpam-1857	112	9	with	with	ADP
ejpam-1857	112	10	(	(	PUNCT
ejpam-1857	112	11	i	i	PROPN
ejpam-1857	112	12	,	,	PUNCT
ejpam-1857	112	13	j	j	PROPN
ejpam-1857	112	14	)	)	PUNCT
ejpam-1857	112	15	=	=	PUNCT
ejpam-1857	112	16	(	(	PUNCT
ejpam-1857	112	17	65	65	NUM
ejpam-1857	112	18	,	,	PUNCT
ejpam-1857	112	19	58	58	NUM
ejpam-1857	112	20	)	)	PUNCT
ejpam-1857	112	21	and	and	CCONJ
ejpam-1857	112	22	its	its	PRON
ejpam-1857	112	23	generators	generator	NOUN
ejpam-1857	112	24	are	be	AUX
ejpam-1857	112	25	the	the	DET
ejpam-1857	112	26	following	follow	VERB
ejpam-1857	112	27	:	:	PUNCT
ejpam-1857	112	28	p1	p1	NOUN
ejpam-1857	112	29	=[	=[	NOUN
ejpam-1857	112	30	57564577194761/1008016,29006793653594700125/1012048064	57564577194761/1008016,29006793653594700125/1012048064	NOUN
ejpam-1857	112	31	]	]	PUNCT
ejpam-1857	112	32	,	,	PUNCT
ejpam-1857	112	33	p2	p2	PROPN
ejpam-1857	112	34	=[	=[	NOUN
ejpam-1857	112	35	165532287616200/2745649,505394258095121556600/4549540393	165532287616200/2745649,505394258095121556600/4549540393	NOUN
ejpam-1857	112	36	]	]	PUNCT
ejpam-1857	112	37	,	,	PUNCT
ejpam-1857	112	38	p3	p3	PROPN
ejpam-1857	112	39	=[	=[	NOUN
ejpam-1857	112	40	6192906993/64,311795186829399/512	6192906993/64,311795186829399/512	PROPN
ejpam-1857	112	41	]	]	X
ejpam-1857	112	42	,	,	PUNCT
ejpam-1857	112	43	p4	p4	ADJ
ejpam-1857	112	44	=[	=[	NOUN
ejpam-1857	112	45	24834332880/121,3321719539155360/1331	24834332880/121,3321719539155360/1331	PROPN
ejpam-1857	112	46	]	]	X
ejpam-1857	112	47	,	,	PUNCT
ejpam-1857	112	48	p5	p5	PROPN
ejpam-1857	112	49	=[	=[	NOUN
ejpam-1857	112	50	341015696,5742307020800	341015696,5742307020800	PROPN
ejpam-1857	112	51	]	]	X
ejpam-1857	112	52	.	.	PUNCT
ejpam-1857	113	1	f.	f.	PROPN
ejpam-1857	113	2	izadi	izadi	PROPN
ejpam-1857	113	3	,	,	PUNCT
ejpam-1857	113	4	k.	k.	PROPN
ejpam-1857	113	5	nabardi	nabardi	PROPN
ejpam-1857	113	6	/	/	SYM
ejpam-1857	113	7	eur	eur	PROPN
ejpam-1857	113	8	.	.	PUNCT
ejpam-1857	114	1	j.	j.	PROPN
ejpam-1857	114	2	pure	pure	PROPN
ejpam-1857	114	3	appl	appl	PROPN
ejpam-1857	114	4	.	.	PROPN
ejpam-1857	114	5	math	math	PROPN
ejpam-1857	114	6	,	,	PUNCT
ejpam-1857	114	7	7	7	NUM
ejpam-1857	114	8	(	(	PUNCT
ejpam-1857	114	9	2014	2014	NUM
ejpam-1857	114	10	)	)	PUNCT
ejpam-1857	114	11	,	,	PUNCT
ejpam-1857	114	12	131	131	NUM
ejpam-1857	114	13	-	-	SYM
ejpam-1857	114	14	139	139	NUM
ejpam-1857	114	15	135	135	NUM
ejpam-1857	114	16	3	3	NUM
ejpam-1857	114	17	.	.	PUNCT
ejpam-1857	114	18	subfamily	subfamily	ADV
ejpam-1857	114	19	of	of	ADP
ejpam-1857	114	20	rank	rank	NOUN
ejpam-1857	114	21	≥	≥	NOUN
ejpam-1857	114	22	2	2	NUM
ejpam-1857	114	23	in	in	ADP
ejpam-1857	114	24	this	this	DET
ejpam-1857	114	25	section	section	NOUN
ejpam-1857	114	26	,	,	PUNCT
ejpam-1857	114	27	we	we	PRON
ejpam-1857	114	28	consider	consider	VERB
ejpam-1857	114	29	a	a	DET
ejpam-1857	114	30	subfamily	subfamily	NOUN
ejpam-1857	114	31	of	of	ADP
ejpam-1857	114	32	(	(	PUNCT
ejpam-1857	114	33	5	5	X
ejpam-1857	114	34	)	)	PUNCT
ejpam-1857	114	35	having	having	AUX
ejpam-1857	114	36	rank	rank	VERB
ejpam-1857	114	37	≥	≥	NOUN
ejpam-1857	114	38	2	2	NUM
ejpam-1857	114	39	.	.	PUNCT
ejpam-1857	114	40	to	to	PART
ejpam-1857	114	41	do	do	VERB
ejpam-1857	114	42	this	this	PRON
ejpam-1857	114	43	,	,	PUNCT
ejpam-1857	114	44	let	let	VERB
ejpam-1857	114	45	the	the	DET
ejpam-1857	114	46	point	point	NOUN
ejpam-1857	114	47	r=	r=	ADJ
ejpam-1857	114	48	(	(	PUNCT
ejpam-1857	114	49	2a2	2a2	NUM
ejpam-1857	114	50	,	,	PUNCT
ejpam-1857	114	51	y0	y0	NOUN
ejpam-1857	114	52	)	)	PUNCT
ejpam-1857	114	53	be	be	AUX
ejpam-1857	114	54	on	on	ADP
ejpam-1857	114	55	(	(	PUNCT
ejpam-1857	114	56	5	5	NUM
ejpam-1857	114	57	)	)	PUNCT
ejpam-1857	114	58	.	.	PUNCT
ejpam-1857	115	1	we	we	PRON
ejpam-1857	115	2	are	be	AUX
ejpam-1857	115	3	going	go	VERB
ejpam-1857	115	4	to	to	PART
ejpam-1857	115	5	show	show	VERB
ejpam-1857	115	6	that	that	SCONJ
ejpam-1857	115	7	the	the	DET
ejpam-1857	115	8	points	point	NOUN
ejpam-1857	115	9	q	q	NOUN
ejpam-1857	115	10	and	and	CCONJ
ejpam-1857	115	11	r	r	NOUN
ejpam-1857	115	12	are	be	AUX
ejpam-1857	115	13	independent	independent	ADJ
ejpam-1857	115	14	.	.	PUNCT
ejpam-1857	116	1	if	if	SCONJ
ejpam-1857	116	2	r=	r=	PROPN
ejpam-1857	116	3	(	(	PUNCT
ejpam-1857	116	4	2a2	2a2	NUM
ejpam-1857	116	5	,	,	PUNCT
ejpam-1857	116	6	y0	y0	NOUN
ejpam-1857	116	7	)	)	PUNCT
ejpam-1857	116	8	be	be	AUX
ejpam-1857	116	9	on	on	ADP
ejpam-1857	116	10	(	(	PUNCT
ejpam-1857	116	11	5	5	NUM
ejpam-1857	116	12	)	)	PUNCT
ejpam-1857	116	13	,	,	PUNCT
ejpam-1857	116	14	then	then	ADV
ejpam-1857	116	15	we	we	PRON
ejpam-1857	116	16	have	have	AUX
ejpam-1857	116	17	y2	y2	VERB
ejpam-1857	116	18	0	0	NUM
ejpam-1857	117	1	=	=	SYM
ejpam-1857	117	2	2a4(2a2	2a4(2a2	NUM
ejpam-1857	117	3	−	−	PROPN
ejpam-1857	117	4	b2	b2	NOUN
ejpam-1857	117	5	)	)	PUNCT
ejpam-1857	117	6	.	.	PUNCT
ejpam-1857	118	1	this	this	PRON
ejpam-1857	118	2	implies	imply	VERB
ejpam-1857	118	3	that	that	SCONJ
ejpam-1857	118	4	2a2	2a2	NUM
ejpam-1857	118	5	−	−	PROPN
ejpam-1857	118	6	b2	b2	NOUN
ejpam-1857	118	7	=	=	SYM
ejpam-1857	118	8	2k2	2k2	NUM
ejpam-1857	118	9	(	(	PUNCT
ejpam-1857	118	10	k	k	PROPN
ejpam-1857	118	11	∈	∈	PROPN
ejpam-1857	118	12	z	z	PROPN
ejpam-1857	118	13	)	)	PUNCT
ejpam-1857	118	14	or	or	CCONJ
ejpam-1857	118	15	equivalently	equivalently	ADV
ejpam-1857	118	16	u4	u4	PROPN
ejpam-1857	118	17	−	−	PROPN
ejpam-1857	118	18	4u2	4u2	NUM
ejpam-1857	119	1	+	+	CCONJ
ejpam-1857	119	2	1=	1=	NUM
ejpam-1857	119	3	v2	v2	NOUN
ejpam-1857	119	4	,	,	PUNCT
ejpam-1857	119	5	(	(	PUNCT
ejpam-1857	119	6	6	6	NUM
ejpam-1857	119	7	)	)	PUNCT
ejpam-1857	119	8	where	where	SCONJ
ejpam-1857	119	9	u=	u=	NOUN
ejpam-1857	119	10	i/	i/	VERB
ejpam-1857	119	11	j	j	PROPN
ejpam-1857	119	12	(	(	PUNCT
ejpam-1857	119	13	or	or	CCONJ
ejpam-1857	119	14	u=	u=	PROPN
ejpam-1857	119	15	j	j	PROPN
ejpam-1857	119	16	/	/	SYM
ejpam-1857	119	17	i	i	PROPN
ejpam-1857	119	18	)	)	PUNCT
ejpam-1857	119	19	and	and	CCONJ
ejpam-1857	119	20	v	v	X
ejpam-1857	119	21	=	=	PUNCT
ejpam-1857	119	22	k/	k/	NOUN
ejpam-1857	119	23	j2	j2	NOUN
ejpam-1857	119	24	(	(	PUNCT
ejpam-1857	119	25	or	or	CCONJ
ejpam-1857	119	26	v	v	NOUN
ejpam-1857	119	27	=	=	PUNCT
ejpam-1857	119	28	k/	k/	NOUN
ejpam-1857	119	29	j2	j2	PROPN
ejpam-1857	119	30	)	)	PUNCT
ejpam-1857	119	31	.	.	PUNCT
ejpam-1857	120	1	from	from	ADP
ejpam-1857	120	2	(	(	PUNCT
ejpam-1857	120	3	6	6	NUM
ejpam-1857	120	4	)	)	PUNCT
ejpam-1857	120	5	one	one	NOUN
ejpam-1857	120	6	can	can	AUX
ejpam-1857	120	7	easily	easily	ADV
ejpam-1857	120	8	get	get	VERB
ejpam-1857	120	9	an	an	DET
ejpam-1857	120	10	elliptic	elliptic	ADJ
ejpam-1857	120	11	curve	curve	NOUN
ejpam-1857	120	12	of	of	ADP
ejpam-1857	120	13	the	the	DET
ejpam-1857	120	14	form	form	NOUN
ejpam-1857	120	15	y2	y2	NOUN
ejpam-1857	120	16	=	=	PUNCT
ejpam-1857	121	1	x3	x3	ADJ
ejpam-1857	121	2	−	−	PROPN
ejpam-1857	122	1	4x2	4x2	NUM
ejpam-1857	123	1	−	−	PROPN
ejpam-1857	123	2	4x	4x	NOUN
ejpam-1857	124	1	+	+	CCONJ
ejpam-1857	125	1	16=	16=	NUM
ejpam-1857	125	2	(	(	PUNCT
ejpam-1857	125	3	x	x	X
ejpam-1857	125	4	+	+	NUM
ejpam-1857	125	5	2)(x	2)(x	NUM
ejpam-1857	125	6	−	−	NOUN
ejpam-1857	125	7	2)(x	2)(x	NUM
ejpam-1857	125	8	−	−	NOUN
ejpam-1857	125	9	4	4	NUM
ejpam-1857	125	10	)	)	PUNCT
ejpam-1857	125	11	.	.	PUNCT
ejpam-1857	126	1	(	(	PUNCT
ejpam-1857	126	2	7	7	X
ejpam-1857	126	3	)	)	PUNCT
ejpam-1857	126	4	this	this	DET
ejpam-1857	126	5	elliptic	elliptic	ADJ
ejpam-1857	126	6	curve	curve	NOUN
ejpam-1857	126	7	has	have	VERB
ejpam-1857	126	8	a	a	DET
ejpam-1857	126	9	group	group	NOUN
ejpam-1857	126	10	of	of	ADP
ejpam-1857	126	11	rational	rational	ADJ
ejpam-1857	126	12	points	point	NOUN
ejpam-1857	126	13	isomorphic	isomorphic	ADJ
ejpam-1857	126	14	to	to	ADP
ejpam-1857	126	15	z×	z×	NUM
ejpam-1857	126	16	(	(	PUNCT
ejpam-1857	126	17	z/2z)×	z/2z)×	X
ejpam-1857	126	18	(	(	PUNCT
ejpam-1857	126	19	z/2z	z/2z	NUM
ejpam-1857	126	20	)	)	PUNCT
ejpam-1857	126	21	.	.	PUNCT
ejpam-1857	127	1	the	the	DET
ejpam-1857	127	2	group	group	NOUN
ejpam-1857	127	3	of	of	ADP
ejpam-1857	127	4	rational	rational	ADJ
ejpam-1857	127	5	points	point	NOUN
ejpam-1857	127	6	is	be	AUX
ejpam-1857	127	7	generated	generate	VERB
ejpam-1857	127	8	by	by	ADP
ejpam-1857	127	9	the	the	DET
ejpam-1857	127	10	torsion	torsion	NOUN
ejpam-1857	127	11	points	point	NOUN
ejpam-1857	127	12	(	(	PUNCT
ejpam-1857	127	13	±2	±2	NOUN
ejpam-1857	127	14	,	,	PUNCT
ejpam-1857	127	15	0	0	NUM
ejpam-1857	127	16	)	)	PUNCT
ejpam-1857	127	17	together	together	ADV
ejpam-1857	127	18	with	with	ADP
ejpam-1857	127	19	the	the	DET
ejpam-1857	127	20	point	point	NOUN
ejpam-1857	127	21	of	of	ADP
ejpam-1857	127	22	infinite	infinite	ADJ
ejpam-1857	127	23	order	order	NOUN
ejpam-1857	127	24	t	t	NOUN
ejpam-1857	127	25	=	=	SYM
ejpam-1857	127	26	(	(	PUNCT
ejpam-1857	127	27	0	0	NUM
ejpam-1857	127	28	,	,	PUNCT
ejpam-1857	127	29	4	4	NUM
ejpam-1857	127	30	)	)	PUNCT
ejpam-1857	127	31	.	.	PUNCT
ejpam-1857	128	1	for	for	ADP
ejpam-1857	128	2	each	each	DET
ejpam-1857	128	3	n	n	PRON
ejpam-1857	128	4	∈	∈	NOUN
ejpam-1857	128	5	n	n	CCONJ
ejpam-1857	128	6	we	we	PRON
ejpam-1857	128	7	have	have	VERB
ejpam-1857	128	8	point	point	NOUN
ejpam-1857	128	9	(	(	PUNCT
ejpam-1857	128	10	xn	xn	PROPN
ejpam-1857	128	11	,	,	PUNCT
ejpam-1857	128	12	yn	yn	PROPN
ejpam-1857	128	13	)	)	PUNCT
ejpam-1857	129	1	=	=	SYM
ejpam-1857	129	2	nt	not	PART
ejpam-1857	129	3	on	on	ADV
ejpam-1857	129	4	(	(	PUNCT
ejpam-1857	129	5	7	7	NUM
ejpam-1857	129	6	)	)	PUNCT
ejpam-1857	129	7	,	,	PUNCT
ejpam-1857	129	8	which	which	PRON
ejpam-1857	129	9	corresponds	correspond	VERB
ejpam-1857	129	10	to	to	ADP
ejpam-1857	129	11	a	a	DET
ejpam-1857	129	12	point	point	NOUN
ejpam-1857	129	13	(	(	PUNCT
ejpam-1857	129	14	un	un	PROPN
ejpam-1857	129	15	,	,	PUNCT
ejpam-1857	129	16	vn	vn	NOUN
ejpam-1857	129	17	)	)	PUNCT
ejpam-1857	129	18	on	on	ADP
ejpam-1857	129	19	(	(	PUNCT
ejpam-1857	129	20	6	6	NUM
ejpam-1857	129	21	)	)	PUNCT
ejpam-1857	129	22	(	(	PUNCT
ejpam-1857	129	23	see	see	VERB
ejpam-1857	129	24	[	[	X
ejpam-1857	129	25	10	10	NUM
ejpam-1857	129	26	,	,	PUNCT
ejpam-1857	129	27	section	section	NOUN
ejpam-1857	129	28	2.17	2.17	NUM
ejpam-1857	129	29	,	,	PUNCT
ejpam-1857	129	30	p.	p.	NOUN
ejpam-1857	129	31	37	37	NUM
ejpam-1857	129	32	]	]	PUNCT
ejpam-1857	129	33	)	)	PUNCT
ejpam-1857	129	34	by	by	ADP
ejpam-1857	129	35	un	un	PROPN
ejpam-1857	129	36	=	=	PROPN
ejpam-1857	129	37	±2(xn	±2(xn	X
ejpam-1857	129	38	−	−	PROPN
ejpam-1857	129	39	4	4	X
ejpam-1857	129	40	)	)	PUNCT
ejpam-1857	129	41	yn	yn	PROPN
ejpam-1857	129	42	,	,	PUNCT
ejpam-1857	129	43	vn	vn	PROPN
ejpam-1857	129	44	=	=	PROPN
ejpam-1857	130	1	∓(2−	∓(2−	PROPN
ejpam-1857	130	2	u2	u2	PROPN
ejpam-1857	130	3	n	n	PROPN
ejpam-1857	130	4	xn	xn	NUM
ejpam-1857	130	5	)	)	PUNCT
ejpam-1857	130	6	2	2	NUM
ejpam-1857	130	7	.	.	PUNCT
ejpam-1857	131	1	(	(	PUNCT
ejpam-1857	131	2	8)	8)	NUM
ejpam-1857	131	3	this	this	DET
ejpam-1857	131	4	turn	turn	NOUN
ejpam-1857	131	5	in	in	ADP
ejpam-1857	131	6	gives	give	VERB
ejpam-1857	131	7	rise	rise	VERB
ejpam-1857	131	8	to	to	ADP
ejpam-1857	131	9	some	some	DET
ejpam-1857	131	10	particular	particular	ADJ
ejpam-1857	131	11	values	value	NOUN
ejpam-1857	131	12	of	of	ADP
ejpam-1857	131	13	(	(	PUNCT
ejpam-1857	131	14	i	i	PROPN
ejpam-1857	131	15	,	,	PUNCT
ejpam-1857	131	16	j	j	PROPN
ejpam-1857	131	17	,	,	PUNCT
ejpam-1857	131	18	k	k	PROPN
ejpam-1857	131	19	)	)	PUNCT
ejpam-1857	131	20	two	two	NUM
ejpam-1857	131	21	points	point	NOUN
ejpam-1857	131	22	of	of	ADP
ejpam-1857	131	23	the	the	DET
ejpam-1857	131	24	forms	form	NOUN
ejpam-1857	131	25	(	(	PUNCT
ejpam-1857	131	26	c2	c2	PROPN
ejpam-1857	131	27	,	,	PUNCT
ejpam-1857	131	28	abc	abc	PROPN
ejpam-1857	131	29	)	)	PUNCT
ejpam-1857	131	30	and	and	CCONJ
ejpam-1857	131	31	(	(	PUNCT
ejpam-1857	131	32	2a2	2a2	NUM
ejpam-1857	131	33	,	,	PUNCT
ejpam-1857	131	34	2a2k	2a2k	NUM
ejpam-1857	131	35	)	)	PUNCT
ejpam-1857	131	36	on	on	ADP
ejpam-1857	131	37	(	(	PUNCT
ejpam-1857	131	38	5	5	NUM
ejpam-1857	131	39	)	)	PUNCT
ejpam-1857	131	40	where	where	SCONJ
ejpam-1857	131	41	k	k	PROPN
ejpam-1857	131	42	is	be	AUX
ejpam-1857	131	43	dependent	dependent	ADJ
ejpam-1857	131	44	on	on	ADP
ejpam-1857	131	45	i	i	PRON
ejpam-1857	131	46	,	,	PUNCT
ejpam-1857	131	47	j.	j.	PROPN
ejpam-1857	131	48	in	in	ADP
ejpam-1857	131	49	the	the	DET
ejpam-1857	131	50	next	next	ADJ
ejpam-1857	131	51	step	step	NOUN
ejpam-1857	131	52	,	,	PUNCT
ejpam-1857	131	53	showing	show	VERB
ejpam-1857	131	54	that	that	SCONJ
ejpam-1857	131	55	these	these	DET
ejpam-1857	131	56	two	two	NUM
ejpam-1857	131	57	points	point	NOUN
ejpam-1857	131	58	(	(	PUNCT
ejpam-1857	131	59	(	(	PUNCT
ejpam-1857	131	60	i2	i2	PROPN
ejpam-1857	131	61	+	+	CCONJ
ejpam-1857	131	62	j2)2	j2)2	PROPN
ejpam-1857	131	63	,	,	PUNCT
ejpam-1857	131	64	2i	2i	NOUN
ejpam-1857	131	65	j(i4	j(i4	PROPN
ejpam-1857	131	66	−	−	PROPN
ejpam-1857	131	67	j4	j4	PROPN
ejpam-1857	131	68	)	)	PUNCT
ejpam-1857	131	69	)	)	PUNCT
ejpam-1857	131	70	and	and	CCONJ
ejpam-1857	131	71	(	(	PUNCT
ejpam-1857	131	72	2(i2	2(i2	NUM
ejpam-1857	131	73	−	−	NOUN
ejpam-1857	131	74	j2)2	j2)2	NOUN
ejpam-1857	131	75	,	,	PUNCT
ejpam-1857	131	76	2(i2	2(i2	NUM
ejpam-1857	131	77	−	−	NOUN
ejpam-1857	131	78	j2)2k	j2)2k	NOUN
ejpam-1857	131	79	)	)	PUNCT
ejpam-1857	131	80	are	be	AUX
ejpam-1857	131	81	independent	independent	ADJ
ejpam-1857	131	82	.	.	PUNCT
ejpam-1857	132	1	now	now	ADV
ejpam-1857	132	2	we	we	PRON
ejpam-1857	132	3	are	be	AUX
ejpam-1857	132	4	going	go	VERB
ejpam-1857	132	5	to	to	PART
ejpam-1857	132	6	find	find	VERB
ejpam-1857	132	7	a	a	DET
ejpam-1857	132	8	value	value	NOUN
ejpam-1857	132	9	of	of	ADP
ejpam-1857	132	10	n	n	NUM
ejpam-1857	132	11	and	and	CCONJ
ejpam-1857	132	12	the	the	DET
ejpam-1857	132	13	corresponding	correspond	VERB
ejpam-1857	132	14	(	(	PUNCT
ejpam-1857	132	15	i	i	PROPN
ejpam-1857	132	16	,	,	PUNCT
ejpam-1857	132	17	j	j	PROPN
ejpam-1857	132	18	,	,	PUNCT
ejpam-1857	132	19	k	k	NOUN
ejpam-1857	132	20	)	)	PUNCT
ejpam-1857	132	21	such	such	ADJ
ejpam-1857	132	22	that	that	SCONJ
ejpam-1857	132	23	these	these	DET
ejpam-1857	132	24	two	two	NUM
ejpam-1857	132	25	points	point	NOUN
ejpam-1857	132	26	be	be	AUX
ejpam-1857	132	27	independent	independent	ADJ
ejpam-1857	132	28	.	.	PUNCT
ejpam-1857	133	1	for	for	ADP
ejpam-1857	133	2	n	n	NOUN
ejpam-1857	133	3	=	=	SYM
ejpam-1857	133	4	2	2	NUM
ejpam-1857	133	5	and	and	CCONJ
ejpam-1857	133	6	consequently	consequently	ADV
ejpam-1857	133	7	(	(	PUNCT
ejpam-1857	133	8	i	i	PROPN
ejpam-1857	133	9	,	,	PUNCT
ejpam-1857	133	10	j	j	PROPN
ejpam-1857	133	11	,	,	PUNCT
ejpam-1857	133	12	k	k	PROPN
ejpam-1857	133	13	)	)	PUNCT
ejpam-1857	133	14	=	=	SYM
ejpam-1857	133	15	(	(	PUNCT
ejpam-1857	133	16	15,4	15,4	INTJ
ejpam-1857	133	17	,	,	PUNCT
ejpam-1857	133	18	191	191	NUM
ejpam-1857	133	19	)	)	PUNCT
ejpam-1857	133	20	,	,	PUNCT
ejpam-1857	133	21	we	we	PRON
ejpam-1857	133	22	obtain	obtain	VERB
ejpam-1857	133	23	the	the	DET
ejpam-1857	133	24	elliptic	elliptic	ADJ
ejpam-1857	133	25	curve	curve	NOUN
ejpam-1857	133	26	y2	y2	PROPN
ejpam-1857	134	1	=	=	SYM
ejpam-1857	134	2	x3	x3	ADJ
ejpam-1857	134	3	−	−	NOUN
ejpam-1857	134	4	58081x2	58081x2	NOUN
ejpam-1857	135	1	+	+	CCONJ
ejpam-1857	135	2	629006400x	629006400x	NOUN
ejpam-1857	135	3	,	,	PUNCT
ejpam-1857	135	4	and	and	CCONJ
ejpam-1857	135	5	the	the	DET
ejpam-1857	135	6	points	point	NOUN
ejpam-1857	136	1	p	p	X
ejpam-1857	136	2	=	=	X
ejpam-1857	136	3	(	(	PUNCT
ejpam-1857	136	4	58081	58081	NUM
ejpam-1857	136	5	,	,	PUNCT
ejpam-1857	136	6	6044280	6044280	NUM
ejpam-1857	136	7	)	)	PUNCT
ejpam-1857	136	8	,	,	PUNCT
ejpam-1857	136	9	q	q	NOUN
ejpam-1857	136	10	=	=	SYM
ejpam-1857	136	11	(	(	PUNCT
ejpam-1857	136	12	87362	87362	NUM
ejpam-1857	136	13	,	,	PUNCT
ejpam-1857	136	14	16686142	16686142	NUM
ejpam-1857	136	15	)	)	PUNCT
ejpam-1857	136	16	.	.	PUNCT
ejpam-1857	137	1	by	by	ADP
ejpam-1857	137	2	using	use	VERB
ejpam-1857	137	3	the	the	DET
ejpam-1857	137	4	sage	sage	NOUN
ejpam-1857	137	5	software	software	NOUN
ejpam-1857	137	6	[	[	X
ejpam-1857	137	7	6	6	NUM
ejpam-1857	137	8	]	]	PUNCT
ejpam-1857	137	9	,	,	PUNCT
ejpam-1857	137	10	we	we	PRON
ejpam-1857	137	11	see	see	VERB
ejpam-1857	137	12	that	that	SCONJ
ejpam-1857	137	13	the	the	DET
ejpam-1857	137	14	associated	associated	ADJ
ejpam-1857	137	15	height	height	NOUN
ejpam-1857	137	16	matrix	matrix	NOUN
ejpam-1857	137	17	has	have	VERB
ejpam-1857	137	18	non	non	ADJ
ejpam-1857	137	19	-	-	ADJ
ejpam-1857	137	20	zero	zero	NUM
ejpam-1857	137	21	determinant	determinant	ADJ
ejpam-1857	137	22	50.3755	50.3755	NUM
ejpam-1857	137	23	showing	show	VERB
ejpam-1857	137	24	that	that	SCONJ
ejpam-1857	137	25	the	the	DET
ejpam-1857	137	26	points	point	NOUN
ejpam-1857	137	27	are	be	AUX
ejpam-1857	137	28	independent	independent	ADJ
ejpam-1857	137	29	.	.	PUNCT
ejpam-1857	138	1	so	so	ADV
ejpam-1857	138	2	the	the	DET
ejpam-1857	138	3	specialization	specialization	NOUN
ejpam-1857	138	4	result	result	NOUN
ejpam-1857	138	5	of	of	ADP
ejpam-1857	138	6	silverman	silverman	NOUN
ejpam-1857	138	7	implies	imply	VERB
ejpam-1857	138	8	that	that	SCONJ
ejpam-1857	138	9	for	for	ADP
ejpam-1857	138	10	all	all	DET
ejpam-1857	138	11	but	but	CCONJ
ejpam-1857	138	12	finitely	finitely	ADV
ejpam-1857	138	13	many	many	ADJ
ejpam-1857	138	14	rational	rational	ADJ
ejpam-1857	138	15	numbers	number	NOUN
ejpam-1857	138	16	,	,	PUNCT
ejpam-1857	138	17	the	the	DET
ejpam-1857	138	18	specialized	specialized	ADJ
ejpam-1857	138	19	curve	curve	NOUN
ejpam-1857	138	20	also	also	ADV
ejpam-1857	138	21	has	have	AUX
ejpam-1857	138	22	rank	rank	VERB
ejpam-1857	138	23	at	at	ADV
ejpam-1857	138	24	least	least	ADJ
ejpam-1857	138	25	2	2	NUM
ejpam-1857	138	26	.	.	X
ejpam-1857	139	1	in	in	ADP
ejpam-1857	139	2	the	the	DET
ejpam-1857	139	3	following	following	NOUN
ejpam-1857	139	4	we	we	PRON
ejpam-1857	139	5	have	have	AUX
ejpam-1857	139	6	listed	list	VERB
ejpam-1857	139	7	some	some	DET
ejpam-1857	139	8	curves	curve	NOUN
ejpam-1857	139	9	of	of	ADP
ejpam-1857	139	10	this	this	DET
ejpam-1857	139	11	type	type	NOUN
ejpam-1857	139	12	with	with	ADP
ejpam-1857	139	13	rank	rank	PROPN
ejpam-1857	139	14	≥	≥	NOUN
ejpam-1857	139	15	2	2	NUM
ejpam-1857	139	16	.	.	PUNCT
ejpam-1857	139	17	f.	f.	PROPN
ejpam-1857	139	18	izadi	izadi	PROPN
ejpam-1857	139	19	,	,	PUNCT
ejpam-1857	139	20	k.	k.	PROPN
ejpam-1857	139	21	nabardi	nabardi	PROPN
ejpam-1857	139	22	/	/	SYM
ejpam-1857	139	23	eur	eur	PROPN
ejpam-1857	139	24	.	.	PUNCT
ejpam-1857	140	1	j.	j.	PROPN
ejpam-1857	140	2	pure	pure	PROPN
ejpam-1857	140	3	appl	appl	PROPN
ejpam-1857	140	4	.	.	PROPN
ejpam-1857	140	5	math	math	PROPN
ejpam-1857	140	6	,	,	PUNCT
ejpam-1857	140	7	7	7	NUM
ejpam-1857	140	8	(	(	PUNCT
ejpam-1857	140	9	2014	2014	NUM
ejpam-1857	140	10	)	)	PUNCT
ejpam-1857	140	11	,	,	PUNCT
ejpam-1857	140	12	131	131	NUM
ejpam-1857	140	13	-	-	SYM
ejpam-1857	140	14	139	139	NUM
ejpam-1857	140	15	136	136	NUM
ejpam-1857	140	16	table	table	NOUN
ejpam-1857	140	17	1	1	NUM
ejpam-1857	140	18	:	:	PUNCT
ejpam-1857	140	19	some	some	DET
ejpam-1857	140	20	curves	curve	NOUN
ejpam-1857	140	21	with	with	ADP
ejpam-1857	140	22	rank≥	rank≥	NOUN
ejpam-1857	140	23	2	2	NUM
ejpam-1857	140	24	.	.	PUNCT
ejpam-1857	141	1	nt	not	PART
ejpam-1857	141	2	(	(	PUNCT
ejpam-1857	141	3	i	i	PROPN
ejpam-1857	141	4	,	,	PUNCT
ejpam-1857	141	5	j	j	PROPN
ejpam-1857	141	6	)	)	PUNCT
ejpam-1857	141	7	curve	curve	NOUN
ejpam-1857	141	8	rank	rank	NOUN
ejpam-1857	141	9	3	3	NUM
ejpam-1857	141	10	t	t	NOUN
ejpam-1857	141	11	(	(	PUNCT
ejpam-1857	141	12	442	442	NUM
ejpam-1857	141	13	,	,	PUNCT
ejpam-1857	141	14	161	161	NUM
ejpam-1857	141	15	)	)	PUNCT
ejpam-1857	142	1	y2	y2	NOUN
ejpam-1857	142	2	=	=	SYM
ejpam-1857	143	1	x3	x3	NUM
ejpam-1857	143	2	−	−	PROPN
ejpam-1857	144	1	48967051225x2	48967051225x2	NOUN
ejpam-1857	145	1	+581572076457241803024x	+581572076457241803024x	PROPN
ejpam-1857	145	2	2≤	2≤	NUM
ejpam-1857	145	3	rank	rank	VERB
ejpam-1857	145	4	≤	≤	NUM
ejpam-1857	145	5	4	4	NUM
ejpam-1857	145	6	4	4	NUM
ejpam-1857	145	7	t	t	NOUN
ejpam-1857	145	8	(	(	PUNCT
ejpam-1857	145	9	50369,22920	50369,22920	ADV
ejpam-1857	145	10	)	)	PUNCT
ejpam-1857	145	11	y2	y2	NOUN
ejpam-1857	145	12	=	=	PUNCT
ejpam-1857	146	1	x3	x3	ADJ
ejpam-1857	146	2	−	−	NOUN
ejpam-1857	147	1	9378064455014478721x2	9378064455014478721x2	NOUN
ejpam-1857	147	2	+21574787239992360293550097486811193600x	+21574787239992360293550097486811193600x	NOUN
ejpam-1857	147	3	2≤	2≤	NUM
ejpam-1857	147	4	rank	rank	VERB
ejpam-1857	147	5	≤	≤	NUM
ejpam-1857	147	6	3	3	NUM
ejpam-1857	147	7	5	5	NUM
ejpam-1857	147	8	t	t	PROPN
ejpam-1857	147	9	(	(	PUNCT
ejpam-1857	147	10	21771082,2706401	21771082,2706401	NUM
ejpam-1857	147	11	)	)	PUNCT
ejpam-1857	147	12	y2	y2	NOUN
ejpam-1857	147	13	=	=	SYM
ejpam-1857	148	1	x3	x3	ADJ
ejpam-1857	148	2	−	−	PROPN
ejpam-1857	148	3	231654135138249459108043425625x2	231654135138249459108043425625x2	NOUN
ejpam-1857	148	4	+3024105303624698175500634675177804	+3024105303624698175500634675177804	NOUN
ejpam-1857	148	5	2≤	2≤	NUM
ejpam-1857	148	6	rank	rank	VERB
ejpam-1857	148	7	≤	≤	NUM
ejpam-1857	148	8	4	4	NUM
ejpam-1857	148	9	1508758565300431760173584x	1508758565300431760173584x	NUM
ejpam-1857	148	10	4	4	NUM
ejpam-1857	148	11	.	.	PUNCT
ejpam-1857	148	12	general	general	ADJ
ejpam-1857	148	13	case	case	NOUN
ejpam-1857	148	14	we	we	PRON
ejpam-1857	148	15	began	begin	VERB
ejpam-1857	148	16	the	the	DET
ejpam-1857	148	17	paper	paper	NOUN
ejpam-1857	148	18	with	with	ADP
ejpam-1857	148	19	the	the	DET
ejpam-1857	148	20	equation	equation	NOUN
ejpam-1857	148	21	a2	a2	PROPN
ejpam-1857	148	22	+	+	CCONJ
ejpam-1857	148	23	b2	b2	NOUN
ejpam-1857	148	24	=	=	PROPN
ejpam-1857	148	25	c2	c2	PROPN
ejpam-1857	148	26	,	,	PUNCT
ejpam-1857	148	27	which	which	PRON
ejpam-1857	148	28	can	can	AUX
ejpam-1857	148	29	be	be	AUX
ejpam-1857	148	30	regarded	regard	VERB
ejpam-1857	148	31	as	as	ADP
ejpam-1857	148	32	the	the	DET
ejpam-1857	148	33	defining	define	VERB
ejpam-1857	148	34	equation	equation	NOUN
ejpam-1857	148	35	of	of	ADP
ejpam-1857	148	36	a	a	DET
ejpam-1857	148	37	rational	rational	ADJ
ejpam-1857	148	38	curve	curve	NOUN
ejpam-1857	148	39	b	b	PROPN
ejpam-1857	148	40	in	in	ADP
ejpam-1857	148	41	p2	p2	NOUN
ejpam-1857	148	42	,	,	PUNCT
ejpam-1857	148	43	defined	define	VERB
ejpam-1857	148	44	over	over	ADP
ejpam-1857	148	45	q.	q.	NOUN
ejpam-1857	148	46	the	the	DET
ejpam-1857	148	47	curve	curve	NOUN
ejpam-1857	148	48	b	b	PROPN
ejpam-1857	148	49	parameterizes	parameterize	VERB
ejpam-1857	148	50	a	a	DET
ejpam-1857	148	51	family	family	NOUN
ejpam-1857	148	52	of	of	ADP
ejpam-1857	148	53	cubic	cubic	ADJ
ejpam-1857	148	54	curves	curve	NOUN
ejpam-1857	148	55	,	,	PUNCT
ejpam-1857	148	56	given	give	VERB
ejpam-1857	148	57	as	as	ADP
ejpam-1857	148	58	y2	y2	PROPN
ejpam-1857	148	59	=	=	SYM
ejpam-1857	148	60	x(x	x(x	PROPN
ejpam-1857	148	61	−	−	PROPN
ejpam-1857	148	62	a2)(x	a2)(x	PROPN
ejpam-1857	148	63	−	−	PROPN
ejpam-1857	148	64	b2	b2	NOUN
ejpam-1857	148	65	)	)	PUNCT
ejpam-1857	148	66	.	.	PUNCT
ejpam-1857	149	1	these	these	DET
ejpam-1857	149	2	curves	curve	NOUN
ejpam-1857	149	3	are	be	AUX
ejpam-1857	149	4	elliptic	elliptic	ADJ
ejpam-1857	149	5	,	,	PUNCT
ejpam-1857	149	6	except	except	SCONJ
ejpam-1857	149	7	over	over	ADP
ejpam-1857	149	8	8	8	NUM
ejpam-1857	149	9	points	point	NOUN
ejpam-1857	149	10	of	of	ADP
ejpam-1857	149	11	b	b	NOUN
ejpam-1857	149	12	,	,	PUNCT
ejpam-1857	149	13	namely	namely	ADV
ejpam-1857	149	14	over	over	ADV
ejpam-1857	149	15	(	(	PUNCT
ejpam-1857	149	16	1	1	NUM
ejpam-1857	149	17	:	:	PUNCT
ejpam-1857	149	18	±1	±1	VERB
ejpam-1857	149	19	:	:	PUNCT
ejpam-1857	149	20	±	±	NUM
ejpam-1857	149	21	p	p	NOUN
ejpam-1857	149	22	2	2	NUM
ejpam-1857	149	23	)	)	PUNCT
ejpam-1857	149	24	and	and	CCONJ
ejpam-1857	149	25	(	(	PUNCT
ejpam-1857	149	26	0	0	NUM
ejpam-1857	149	27	:	:	SYM
ejpam-1857	149	28	1	1	NUM
ejpam-1857	149	29	:	:	PUNCT
ejpam-1857	149	30	±1	±1	VERB
ejpam-1857	149	31	)	)	PUNCT
ejpam-1857	149	32	and	and	CCONJ
ejpam-1857	149	33	(	(	PUNCT
ejpam-1857	149	34	1	1	NUM
ejpam-1857	149	35	:	:	SYM
ejpam-1857	149	36	0	0	NUM
ejpam-1857	149	37	:	:	PUNCT
ejpam-1857	149	38	±1	±1	PROPN
ejpam-1857	149	39	)	)	PUNCT
ejpam-1857	149	40	.	.	PUNCT
ejpam-1857	150	1	the	the	DET
ejpam-1857	150	2	curve	curve	PROPN
ejpam-1857	150	3	b	b	PROPN
ejpam-1857	150	4	is	be	AUX
ejpam-1857	150	5	isomorphic	isomorphic	ADJ
ejpam-1857	150	6	to	to	ADP
ejpam-1857	150	7	p1	p1	NOUN
ejpam-1857	150	8	;	;	PUNCT
ejpam-1857	150	9	the	the	DET
ejpam-1857	150	10	isomorphism	isomorphism	NOUN
ejpam-1857	150	11	is	be	AUX
ejpam-1857	150	12	presented	present	VERB
ejpam-1857	150	13	in	in	ADP
ejpam-1857	150	14	(	(	PUNCT
ejpam-1857	150	15	4	4	NUM
ejpam-1857	150	16	):	):	PUNCT
ejpam-1857	150	17	p1	p1	PROPN
ejpam-1857	150	18	'	'	PUNCT
ejpam-1857	150	19	b	b	NOUN
ejpam-1857	150	20	:	:	PUNCT
ejpam-1857	150	21	(	(	PUNCT
ejpam-1857	150	22	i	i	PRON
ejpam-1857	150	23	:	:	PUNCT
ejpam-1857	150	24	j	j	X
ejpam-1857	150	25	)	)	PUNCT
ejpam-1857	150	26	7−→	7−→	PROPN
ejpam-1857	150	27	(	(	PUNCT
ejpam-1857	150	28	i2	i2	PROPN
ejpam-1857	150	29	−	−	PROPN
ejpam-1857	150	30	j2	j2	PROPN
ejpam-1857	150	31	:	:	PUNCT
ejpam-1857	150	32	2i	2i	NUM
ejpam-1857	150	33	j	j	PROPN
ejpam-1857	150	34	:	:	PUNCT
ejpam-1857	150	35	i2	i2	PROPN
ejpam-1857	150	36	+	+	CCONJ
ejpam-1857	150	37	j2	j2	PROPN
ejpam-1857	150	38	)	)	PUNCT
ejpam-1857	150	39	.	.	PUNCT
ejpam-1857	151	1	with	with	ADP
ejpam-1857	151	2	t	t	PROPN
ejpam-1857	151	3	:	:	PUNCT
ejpam-1857	151	4	=	=	PUNCT
ejpam-1857	151	5	i/	i/	PROPN
ejpam-1857	151	6	j	j	PROPN
ejpam-1857	151	7	as	as	ADP
ejpam-1857	151	8	a	a	DET
ejpam-1857	151	9	coordinate	coordinate	NOUN
ejpam-1857	151	10	function	function	NOUN
ejpam-1857	151	11	on	on	ADP
ejpam-1857	151	12	p1	p1	PROPN
ejpam-1857	151	13	,	,	PUNCT
ejpam-1857	151	14	this	this	PRON
ejpam-1857	151	15	allows	allow	VERB
ejpam-1857	151	16	to	to	PART
ejpam-1857	151	17	write	write	VERB
ejpam-1857	151	18	the	the	DET
ejpam-1857	151	19	family	family	NOUN
ejpam-1857	151	20	of	of	ADP
ejpam-1857	151	21	cubics	cubic	NOUN
ejpam-1857	151	22	as	as	ADP
ejpam-1857	151	23	y2	y2	PROPN
ejpam-1857	151	24	=	=	SYM
ejpam-1857	151	25	x(x	x(x	PROPN
ejpam-1857	152	1	−	−	PROPN
ejpam-1857	152	2	(	(	PUNCT
ejpam-1857	152	3	t2	t2	NOUN
ejpam-1857	152	4	−	−	PROPN
ejpam-1857	152	5	1)2)(x	1)2)(x	PROPN
ejpam-1857	152	6	−	−	PROPN
ejpam-1857	152	7	4t2	4t2	NUM
ejpam-1857	152	8	)	)	PUNCT
ejpam-1857	152	9	.	.	PUNCT
ejpam-1857	153	1	(	(	PUNCT
ejpam-1857	153	2	9	9	X
ejpam-1857	153	3	)	)	PUNCT
ejpam-1857	153	4	this	this	DET
ejpam-1857	153	5	equation	equation	NOUN
ejpam-1857	153	6	corresponds	correspond	VERB
ejpam-1857	153	7	to	to	ADP
ejpam-1857	153	8	an	an	DET
ejpam-1857	153	9	elliptic	elliptic	ADJ
ejpam-1857	153	10	surface	surface	NOUN
ejpam-1857	153	11	π	π	NOUN
ejpam-1857	153	12	:	:	PUNCT
ejpam-1857	153	13	e	e	X
ejpam-1857	153	14	→	→	SYM
ejpam-1857	153	15	p1	p1	PROPN
ejpam-1857	153	16	defined	define	VERB
ejpam-1857	153	17	over	over	ADP
ejpam-1857	153	18	q	q	NOUN
ejpam-1857	153	19	,	,	PUNCT
ejpam-1857	153	20	or	or	CCONJ
ejpam-1857	153	21	equivalently	equivalently	ADV
ejpam-1857	153	22	,	,	PUNCT
ejpam-1857	153	23	an	an	DET
ejpam-1857	153	24	elliptic	elliptic	ADJ
ejpam-1857	153	25	curve	curve	NOUN
ejpam-1857	153	26	e	e	NOUN
ejpam-1857	153	27	over	over	ADP
ejpam-1857	153	28	the	the	DET
ejpam-1857	153	29	rational	rational	ADJ
ejpam-1857	153	30	function	function	NOUN
ejpam-1857	153	31	field	field	NOUN
ejpam-1857	153	32	q(t	q(t	PROPN
ejpam-1857	153	33	)	)	PUNCT
ejpam-1857	153	34	.	.	PUNCT
ejpam-1857	154	1	the	the	DET
ejpam-1857	154	2	surface	surface	NOUN
ejpam-1857	154	3	e	e	NOUN
ejpam-1857	154	4	is	be	AUX
ejpam-1857	154	5	a	a	DET
ejpam-1857	154	6	so	so	ADV
ejpam-1857	154	7	-	-	PUNCT
ejpam-1857	154	8	called	call	VERB
ejpam-1857	154	9	k3surface	k3surface	NOUN
ejpam-1857	154	10	.	.	PUNCT
ejpam-1857	155	1	from	from	ADP
ejpam-1857	155	2	tate	tate	PROPN
ejpam-1857	155	3	’s	’s	PART
ejpam-1857	155	4	algorithm	algorithm	NOUN
ejpam-1857	155	5	one	one	NUM
ejpam-1857	155	6	calculates	calculate	VERB
ejpam-1857	155	7	that	that	SCONJ
ejpam-1857	155	8	of	of	ADP
ejpam-1857	155	9	the	the	DET
ejpam-1857	155	10	8	8	NUM
ejpam-1857	155	11	singular	singular	ADJ
ejpam-1857	155	12	fibers	fiber	NOUN
ejpam-1857	155	13	of	of	ADP
ejpam-1857	155	14	π	π	PROPN
ejpam-1857	155	15	,	,	PUNCT
ejpam-1857	155	16	4	4	NUM
ejpam-1857	155	17	are	be	AUX
ejpam-1857	155	18	of	of	ADP
ejpam-1857	155	19	type	type	NOUN
ejpam-1857	155	20	i4	i4	PROPN
ejpam-1857	155	21	and	and	CCONJ
ejpam-1857	155	22	the	the	DET
ejpam-1857	155	23	other	other	ADJ
ejpam-1857	155	24	4	4	NUM
ejpam-1857	155	25	are	be	AUX
ejpam-1857	155	26	of	of	ADP
ejpam-1857	155	27	type	type	NOUN
ejpam-1857	155	28	i2	i2	PROPN
ejpam-1857	155	29	.	.	PUNCT
ejpam-1857	156	1	moreover	moreover	ADV
ejpam-1857	156	2	one	one	NOUN
ejpam-1857	156	3	finds	find	VERB
ejpam-1857	156	4	two	two	NUM
ejpam-1857	156	5	points	point	NOUN
ejpam-1857	156	6	in	in	ADP
ejpam-1857	156	7	e(q	e(q	PROPN
ejpam-1857	156	8	[	[	PUNCT
ejpam-1857	156	9	p	p	NOUN
ejpam-1857	156	10	−2](t	−2](t	NOUN
ejpam-1857	156	11	)	)	PUNCT
ejpam-1857	156	12	)	)	PUNCT
ejpam-1857	156	13	namely	namely	ADV
ejpam-1857	156	14	s1	s1	NOUN
ejpam-1857	156	15	:	:	PUNCT
ejpam-1857	156	16	=	=	SYM
ejpam-1857	156	17	(	(	PUNCT
ejpam-1857	156	18	(	(	PUNCT
ejpam-1857	156	19	t2	t2	NOUN
ejpam-1857	156	20	+	+	CCONJ
ejpam-1857	156	21	1)2	1)2	NUM
ejpam-1857	156	22	,	,	PUNCT
ejpam-1857	156	23	2t(t4	2t(t4	NUM
ejpam-1857	156	24	−	−	NOUN
ejpam-1857	156	25	1	1	NUM
ejpam-1857	156	26	)	)	PUNCT
ejpam-1857	156	27	)	)	PUNCT
ejpam-1857	156	28	and	and	CCONJ
ejpam-1857	156	29	s2	s2	VERB
ejpam-1857	156	30	:	:	PUNCT
ejpam-1857	156	31	=	=	SYM
ejpam-1857	156	32	(	(	PUNCT
ejpam-1857	156	33	−4t2	−4t2	NUM
ejpam-1857	156	34	,	,	PUNCT
ejpam-1857	156	35	4t(t	4t(t	PROPN
ejpam-1857	156	36	+	+	CCONJ
ejpam-1857	156	37	1	1	X
ejpam-1857	156	38	)	)	PUNCT
ejpam-1857	156	39	p	p	PRON
ejpam-1857	156	40	−2	−2	NOUN
ejpam-1857	156	41	)	)	PUNCT
ejpam-1857	156	42	.	.	PUNCT
ejpam-1857	157	1	note	note	VERB
ejpam-1857	157	2	that	that	SCONJ
ejpam-1857	157	3	s1	s1	NOUN
ejpam-1857	157	4	specializes	specialize	VERB
ejpam-1857	157	5	to	to	ADP
ejpam-1857	157	6	the	the	DET
ejpam-1857	157	7	point	point	NOUN
ejpam-1857	157	8	q	q	NOUN
ejpam-1857	157	9	used	use	VERB
ejpam-1857	157	10	in	in	ADP
ejpam-1857	157	11	the	the	DET
ejpam-1857	157	12	proof	proof	NOUN
ejpam-1857	157	13	of	of	ADP
ejpam-1857	157	14	the	the	DET
ejpam-1857	157	15	theorem	theorem	NOUN
ejpam-1857	157	16	2	2	NUM
ejpam-1857	157	17	.	.	PUNCT
ejpam-1857	157	18	a	a	DET
ejpam-1857	157	19	standard	standard	ADJ
ejpam-1857	157	20	intersection	intersection	NOUN
ejpam-1857	157	21	calculation	calculation	NOUN
ejpam-1857	157	22	[	[	X
ejpam-1857	157	23	9	9	X
ejpam-1857	157	24	]	]	PUNCT
ejpam-1857	157	25	shows	show	VERB
ejpam-1857	157	26	that	that	SCONJ
ejpam-1857	157	27	s1	s1	NOUN
ejpam-1857	157	28	and	and	CCONJ
ejpam-1857	157	29	s2	s2	NOUN
ejpam-1857	157	30	are	be	AUX
ejpam-1857	157	31	linearly	linearly	ADV
ejpam-1857	157	32	independent	independent	ADJ
ejpam-1857	157	33	.	.	PUNCT
ejpam-1857	158	1	using	use	VERB
ejpam-1857	158	2	the	the	DET
ejpam-1857	158	3	shioda	shioda	NOUN
ejpam-1857	158	4	-	-	PUNCT
ejpam-1857	158	5	tate	tate	NOUN
ejpam-1857	158	6	formula	formula	NOUN
ejpam-1857	158	7	for	for	ADP
ejpam-1857	158	8	π	π	PROPN
ejpam-1857	158	9	:	:	PUNCT
ejpam-1857	158	10	e	e	X
ejpam-1857	158	11	→	→	SYM
ejpam-1857	158	12	p1	p1	PROPN
ejpam-1857	158	13	and	and	CCONJ
ejpam-1857	158	14	the	the	DET
ejpam-1857	158	15	fact	fact	NOUN
ejpam-1857	158	16	that	that	SCONJ
ejpam-1857	158	17	e	e	NOUN
ejpam-1857	158	18	is	be	AUX
ejpam-1857	158	19	k3	k3	ADJ
ejpam-1857	158	20	,	,	PUNCT
ejpam-1857	158	21	it	it	PRON
ejpam-1857	158	22	follows	follow	VERB
ejpam-1857	158	23	that	that	SCONJ
ejpam-1857	158	24	20≥	20≥	NUM
ejpam-1857	158	25	2	2	NUM
ejpam-1857	158	26	+	+	NUM
ejpam-1857	158	27	4	4	NUM
ejpam-1857	158	28	·	·	SYM
ejpam-1857	158	29	3	3	NUM
ejpam-1857	158	30	+	+	NUM
ejpam-1857	158	31	4	4	NUM
ejpam-1857	158	32	·	·	SYM
ejpam-1857	158	33	1	1	NUM
ejpam-1857	159	1	+	+	NUM
ejpam-1857	159	2	rank(e(q	rank(e(q	NOUN
ejpam-1857	159	3	[	[	PUNCT
ejpam-1857	159	4	p	p	X
ejpam-1857	159	5	−2](t)))≥	−2](t)))≥	NOUN
ejpam-1857	159	6	20	20	NUM
ejpam-1857	159	7	,	,	PUNCT
ejpam-1857	159	8	hence	hence	ADV
ejpam-1857	159	9	rank(e(q	rank(e(q	NOUN
ejpam-1857	159	10	[	[	PUNCT
ejpam-1857	159	11	p	p	NOUN
ejpam-1857	159	12	−2](t	−2](t	NOUN
ejpam-1857	159	13	)	)	PUNCT
ejpam-1857	159	14	)	)	PUNCT
ejpam-1857	159	15	)	)	PUNCT
ejpam-1857	160	1	=	=	PUNCT
ejpam-1857	161	1	2	2	X
ejpam-1857	161	2	.	.	PUNCT
ejpam-1857	161	3	the	the	DET
ejpam-1857	161	4	nontrivial	nontrivial	PROPN
ejpam-1857	161	5	element	element	PROPN
ejpam-1857	161	6	σ	σ	PROPN
ejpam-1857	161	7	∈	∈	PROPN
ejpam-1857	161	8	gal(q	gal(q	PROPN
ejpam-1857	161	9	[	[	PUNCT
ejpam-1857	161	10	p	p	NOUN
ejpam-1857	161	11	−2]/q	−2]/q	NOUN
ejpam-1857	161	12	)	)	PUNCT
ejpam-1857	161	13	satisfies	satisfie	NOUN
ejpam-1857	161	14	σ(s1	σ(s1	ADJ
ejpam-1857	161	15	)	)	PUNCT
ejpam-1857	161	16	=	=	SYM
ejpam-1857	161	17	s1	s1	NOUN
ejpam-1857	161	18	and	and	CCONJ
ejpam-1857	161	19	σ(s2	σ(s2	NOUN
ejpam-1857	161	20	)	)	PUNCT
ejpam-1857	162	1	=	=	SYM
ejpam-1857	162	2	−s2	−s2	PROPN
ejpam-1857	162	3	.	.	PUNCT
ejpam-1857	163	1	as	as	ADP
ejpam-1857	163	2	a	a	DET
ejpam-1857	163	3	consequence	consequence	NOUN
ejpam-1857	163	4	,	,	PUNCT
ejpam-1857	163	5	the	the	DET
ejpam-1857	163	6	+1	+1	PROPN
ejpam-1857	163	7	-	-	PUNCT
ejpam-1857	163	8	eigenspace	eigenspace	NOUN
ejpam-1857	163	9	of	of	ADP
ejpam-1857	163	10	σ	σ	PROPN
ejpam-1857	163	11	,	,	PUNCT
ejpam-1857	163	12	i.e.	i.e.	X
ejpam-1857	163	13	,	,	PUNCT
ejpam-1857	163	14	the	the	DET
ejpam-1857	163	15	subgroup	subgroup	NOUN
ejpam-1857	163	16	e(q(t	e(q(t	PROPN
ejpam-1857	163	17	)	)	PUNCT
ejpam-1857	163	18	)	)	PUNCT
ejpam-1857	163	19	,	,	PUNCT
ejpam-1857	163	20	has	have	AUX
ejpam-1857	163	21	rank	rank	NOUN
ejpam-1857	163	22	1	1	NUM
ejpam-1857	163	23	.	.	PUNCT
ejpam-1857	164	1	silverman	silverman	PROPN
ejpam-1857	164	2	’s	’s	PART
ejpam-1857	164	3	specialization	specialization	NOUN
ejpam-1857	164	4	result	result	NOUN
ejpam-1857	164	5	implies	imply	VERB
ejpam-1857	164	6	that	that	SCONJ
ejpam-1857	164	7	for	for	ADP
ejpam-1857	164	8	all	all	DET
ejpam-1857	164	9	but	but	CCONJ
ejpam-1857	164	10	infinitely	infinitely	ADV
ejpam-1857	164	11	many	many	ADJ
ejpam-1857	164	12	specialization	specialization	NOUN
ejpam-1857	164	13	of	of	ADP
ejpam-1857	164	14	t	t	PROPN
ejpam-1857	164	15	to	to	ADP
ejpam-1857	164	16	a	a	DET
ejpam-1857	164	17	rational	rational	ADJ
ejpam-1857	164	18	number	number	NOUN
ejpam-1857	164	19	,	,	PUNCT
ejpam-1857	164	20	the	the	DET
ejpam-1857	164	21	specialized	specialized	ADJ
ejpam-1857	164	22	elliptic	elliptic	ADJ
ejpam-1857	164	23	curve	curve	NOUN
ejpam-1857	164	24	over	over	ADP
ejpam-1857	164	25	q	q	NOUN
ejpam-1857	164	26	obtained	obtain	VERB
ejpam-1857	164	27	in	in	ADP
ejpam-1857	164	28	this	this	DET
ejpam-1857	164	29	way	way	NOUN
ejpam-1857	164	30	has	have	VERB
ejpam-1857	164	31	positive	positive	ADJ
ejpam-1857	164	32	rank	rank	NOUN
ejpam-1857	164	33	as	as	ADV
ejpam-1857	164	34	well	well	ADV
ejpam-1857	164	35	.	.	PUNCT
ejpam-1857	165	1	f.	f.	PROPN
ejpam-1857	165	2	izadi	izadi	PROPN
ejpam-1857	165	3	,	,	PUNCT
ejpam-1857	165	4	k.	k.	PROPN
ejpam-1857	165	5	nabardi	nabardi	PROPN
ejpam-1857	165	6	/	/	SYM
ejpam-1857	165	7	eur	eur	PROPN
ejpam-1857	165	8	.	.	PUNCT
ejpam-1857	166	1	j.	j.	PROPN
ejpam-1857	166	2	pure	pure	PROPN
ejpam-1857	166	3	appl	appl	PROPN
ejpam-1857	166	4	.	.	PROPN
ejpam-1857	166	5	math	math	PROPN
ejpam-1857	166	6	,	,	PUNCT
ejpam-1857	166	7	7	7	NUM
ejpam-1857	166	8	(	(	PUNCT
ejpam-1857	166	9	2014	2014	NUM
ejpam-1857	166	10	)	)	PUNCT
ejpam-1857	166	11	,	,	PUNCT
ejpam-1857	166	12	131	131	NUM
ejpam-1857	166	13	-	-	SYM
ejpam-1857	166	14	139	139	NUM
ejpam-1857	166	15	137	137	NUM
ejpam-1857	166	16	theorem	theorem	NOUN
ejpam-1857	166	17	2	2	NUM
ejpam-1857	166	18	shows	show	VERB
ejpam-1857	166	19	that	that	SCONJ
ejpam-1857	166	20	in	in	ADP
ejpam-1857	166	21	fact	fact	NOUN
ejpam-1857	166	22	a	a	DET
ejpam-1857	166	23	stronger	strong	ADJ
ejpam-1857	166	24	assertion	assertion	NOUN
ejpam-1857	166	25	is	be	AUX
ejpam-1857	166	26	true	true	ADJ
ejpam-1857	166	27	:	:	PUNCT
ejpam-1857	166	28	whenever	whenever	SCONJ
ejpam-1857	166	29	this	this	DET
ejpam-1857	166	30	specialization	specialization	NOUN
ejpam-1857	166	31	defines	define	VERB
ejpam-1857	166	32	an	an	DET
ejpam-1857	166	33	elliptic	elliptic	ADJ
ejpam-1857	166	34	curve	curve	NOUN
ejpam-1857	166	35	(	(	PUNCT
ejpam-1857	166	36	so	so	ADV
ejpam-1857	166	37	,	,	PUNCT
ejpam-1857	166	38	for	for	ADP
ejpam-1857	166	39	all	all	DET
ejpam-1857	166	40	rational	rational	ADJ
ejpam-1857	166	41	t0	t0	PROPN
ejpam-1857	166	42	6=	6=	SYM
ejpam-1857	166	43	{	{	PUNCT
ejpam-1857	166	44	0,±1	0,±1	NOUN
ejpam-1857	166	45	}	}	PUNCT
ejpam-1857	166	46	)	)	PUNCT
ejpam-1857	166	47	,	,	PUNCT
ejpam-1857	166	48	the	the	DET
ejpam-1857	166	49	rank	rank	NOUN
ejpam-1857	166	50	is	be	AUX
ejpam-1857	166	51	positive	positive	ADJ
ejpam-1857	166	52	.	.	PUNCT
ejpam-1857	167	1	it	it	PRON
ejpam-1857	167	2	is	be	AUX
ejpam-1857	167	3	interesting	interesting	ADJ
ejpam-1857	167	4	to	to	PART
ejpam-1857	167	5	remark	remark	VERB
ejpam-1857	167	6	that	that	SCONJ
ejpam-1857	167	7	e	e	NOUN
ejpam-1857	167	8	provides	provide	VERB
ejpam-1857	167	9	an	an	DET
ejpam-1857	167	10	example	example	NOUN
ejpam-1857	167	11	of	of	ADP
ejpam-1857	167	12	a	a	DET
ejpam-1857	167	13	so	so	ADV
ejpam-1857	167	14	-	-	PUNCT
ejpam-1857	167	15	called	call	VERB
ejpam-1857	167	16	singular	singular	ADJ
ejpam-1857	167	17	k3	k3	PROPN
ejpam-1857	167	18	-	-	PUNCT
ejpam-1857	167	19	surface	surface	NOUN
ejpam-1857	167	20	defined	define	VERB
ejpam-1857	167	21	over	over	ADP
ejpam-1857	167	22	q	q	NOUN
ejpam-1857	167	23	:	:	PUNCT
ejpam-1857	167	24	one	one	NUM
ejpam-1857	167	25	such	such	ADJ
ejpam-1857	167	26	that	that	SCONJ
ejpam-1857	167	27	the	the	DET
ejpam-1857	167	28	picard	picard	NOUN
ejpam-1857	167	29	number	number	NOUN
ejpam-1857	167	30	attains	attain	VERB
ejpam-1857	167	31	the	the	DET
ejpam-1857	167	32	maximal	maximal	ADJ
ejpam-1857	167	33	value	value	NOUN
ejpam-1857	167	34	20	20	NUM
ejpam-1857	167	35	.	.	PUNCT
ejpam-1857	168	1	however	however	ADV
ejpam-1857	168	2	,	,	PUNCT
ejpam-1857	168	3	this	this	DET
ejpam-1857	168	4	value	value	NOUN
ejpam-1857	168	5	20	20	NUM
ejpam-1857	168	6	is	be	AUX
ejpam-1857	168	7	only	only	ADV
ejpam-1857	168	8	attained	attain	VERB
ejpam-1857	168	9	over	over	ADP
ejpam-1857	168	10	an	an	DET
ejpam-1857	168	11	extension	extension	NOUN
ejpam-1857	168	12	of	of	ADP
ejpam-1857	168	13	q	q	NOUN
ejpam-1857	168	14	;	;	PUNCT
ejpam-1857	168	15	a	a	DET
ejpam-1857	168	16	small	small	ADJ
ejpam-1857	168	17	calculation	calculation	NOUN
ejpam-1857	168	18	shows	show	VERB
ejpam-1857	168	19	it	it	PRON
ejpam-1857	168	20	is	be	AUX
ejpam-1857	168	21	the	the	DET
ejpam-1857	168	22	extension	extension	NOUN
ejpam-1857	169	1	q	q	X
ejpam-1857	170	1	[	[	PUNCT
ejpam-1857	170	2	p	p	NOUN
ejpam-1857	170	3	2	2	NUM
ejpam-1857	170	4	,	,	PUNCT
ejpam-1857	170	5	p	p	DET
ejpam-1857	170	6	−2	−2	NOUN
ejpam-1857	170	7	]	]	X
ejpam-1857	170	8	.	.	PUNCT
ejpam-1857	171	1	more	more	ADJ
ejpam-1857	171	2	on	on	ADP
ejpam-1857	171	3	such	such	ADJ
ejpam-1857	171	4	singular	singular	NOUN
ejpam-1857	171	5	k3	k3	PROPN
ejpam-1857	171	6	’s	’s	NOUN
ejpam-1857	171	7	over	over	ADP
ejpam-1857	171	8	q	q	NOUN
ejpam-1857	171	9	can	can	AUX
ejpam-1857	171	10	be	be	AUX
ejpam-1857	171	11	read	read	VERB
ejpam-1857	171	12	in	in	ADP
ejpam-1857	171	13	[	[	X
ejpam-1857	171	14	7	7	NUM
ejpam-1857	171	15	]	]	PUNCT
ejpam-1857	171	16	.	.	PUNCT
ejpam-1857	172	1	to	to	PART
ejpam-1857	172	2	obtain	obtain	VERB
ejpam-1857	172	3	larger	large	ADJ
ejpam-1857	172	4	ranks	rank	NOUN
ejpam-1857	172	5	over	over	ADP
ejpam-1857	172	6	q	q	PROPN
ejpam-1857	172	7	one	one	NUM
ejpam-1857	172	8	now	now	ADV
ejpam-1857	172	9	applies	apply	VERB
ejpam-1857	172	10	base	base	NOUN
ejpam-1857	172	11	changes	change	NOUN
ejpam-1857	172	12	.	.	PUNCT
ejpam-1857	173	1	this	this	DET
ejpam-1857	173	2	idea	idea	NOUN
ejpam-1857	173	3	is	be	AUX
ejpam-1857	173	4	well	well	ADV
ejpam-1857	173	5	known	know	VERB
ejpam-1857	173	6	;	;	PUNCT
ejpam-1857	173	7	for	for	ADP
ejpam-1857	173	8	example	example	NOUN
ejpam-1857	173	9	,	,	PUNCT
ejpam-1857	173	10	it	it	PRON
ejpam-1857	173	11	was	be	AUX
ejpam-1857	173	12	used	use	VERB
ejpam-1857	173	13	in	in	ADP
ejpam-1857	173	14	[	[	X
ejpam-1857	173	15	3	3	NUM
ejpam-1857	173	16	]	]	PUNCT
ejpam-1857	173	17	.	.	PUNCT
ejpam-1857	174	1	a	a	DET
ejpam-1857	174	2	simple	simple	ADJ
ejpam-1857	174	3	explanation	explanation	NOUN
ejpam-1857	174	4	is	be	AUX
ejpam-1857	174	5	that	that	SCONJ
ejpam-1857	174	6	one	one	NUM
ejpam-1857	174	7	replace	replace	VERB
ejpam-1857	174	8	the	the	DET
ejpam-1857	174	9	field	field	NOUN
ejpam-1857	174	10	q(t	q(t	NOUN
ejpam-1857	174	11	)	)	PUNCT
ejpam-1857	174	12	by	by	ADP
ejpam-1857	174	13	a	a	DET
ejpam-1857	174	14	finite	finite	ADJ
ejpam-1857	174	15	extension	extension	NOUN
ejpam-1857	174	16	q(c	q(c	PROPN
ejpam-1857	174	17	)	)	PUNCT
ejpam-1857	174	18	,	,	PUNCT
ejpam-1857	174	19	the	the	DET
ejpam-1857	174	20	function	function	NOUN
ejpam-1857	174	21	field	field	NOUN
ejpam-1857	174	22	of	of	ADP
ejpam-1857	174	23	some	some	DET
ejpam-1857	174	24	curve	curve	NOUN
ejpam-1857	174	25	c	c	PROPN
ejpam-1857	174	26	defined	define	VERB
ejpam-1857	174	27	over	over	ADP
ejpam-1857	174	28	q.	q.	NOUN
ejpam-1857	174	29	if	if	SCONJ
ejpam-1857	174	30	e(q(c	e(q(c	ADV
ejpam-1857	174	31	)	)	PUNCT
ejpam-1857	174	32	)	)	PUNCT
ejpam-1857	175	1	has	have	AUX
ejpam-1857	175	2	rank	rank	NOUN
ejpam-1857	175	3	r	r	NOUN
ejpam-1857	175	4	,	,	PUNCT
ejpam-1857	175	5	then	then	ADV
ejpam-1857	175	6	the	the	DET
ejpam-1857	175	7	same	same	ADJ
ejpam-1857	175	8	specialization	specialization	NOUN
ejpam-1857	175	9	result	result	NOUN
ejpam-1857	175	10	of	of	ADP
ejpam-1857	175	11	silverman	silverman	NOUN
ejpam-1857	175	12	implies	imply	VERB
ejpam-1857	175	13	that	that	SCONJ
ejpam-1857	175	14	for	for	ADP
ejpam-1857	175	15	all	all	DET
ejpam-1857	175	16	but	but	CCONJ
ejpam-1857	175	17	only	only	ADV
ejpam-1857	175	18	finitely	finitely	ADV
ejpam-1857	175	19	many	many	ADJ
ejpam-1857	175	20	rational	rational	ADJ
ejpam-1857	175	21	points	point	NOUN
ejpam-1857	175	22	on	on	ADP
ejpam-1857	175	23	c	c	PROPN
ejpam-1857	175	24	,	,	PUNCT
ejpam-1857	175	25	the	the	DET
ejpam-1857	175	26	specialized	specialized	ADJ
ejpam-1857	175	27	curve	curve	NOUN
ejpam-1857	175	28	also	also	ADV
ejpam-1857	175	29	has	have	AUX
ejpam-1857	175	30	rank	rank	NOUN
ejpam-1857	175	31	at	at	ADP
ejpam-1857	175	32	least	least	ADJ
ejpam-1857	175	33	r.	r.	PROPN
ejpam-1857	175	34	in	in	ADP
ejpam-1857	175	35	particular	particular	ADJ
ejpam-1857	175	36	,	,	PUNCT
ejpam-1857	175	37	this	this	PRON
ejpam-1857	175	38	can	can	AUX
ejpam-1857	175	39	only	only	ADV
ejpam-1857	175	40	be	be	AUX
ejpam-1857	175	41	used	use	VERB
ejpam-1857	175	42	for	for	ADP
ejpam-1857	175	43	constructing	construct	VERB
ejpam-1857	175	44	infinitely	infinitely	ADV
ejpam-1857	175	45	many	many	ADJ
ejpam-1857	175	46	specializations	specialization	NOUN
ejpam-1857	175	47	with	with	ADP
ejpam-1857	175	48	high	high	ADJ
ejpam-1857	175	49	rank	rank	NOUN
ejpam-1857	175	50	,	,	PUNCT
ejpam-1857	175	51	of	of	ADP
ejpam-1857	175	52	the	the	DET
ejpam-1857	175	53	curve	curve	NOUN
ejpam-1857	175	54	c	c	PROPN
ejpam-1857	175	55	contains	contain	VERB
ejpam-1857	175	56	infinitely	infinitely	ADV
ejpam-1857	175	57	many	many	ADJ
ejpam-1857	175	58	rational	rational	ADJ
ejpam-1857	175	59	points	point	NOUN
ejpam-1857	175	60	.	.	PUNCT
ejpam-1857	176	1	if	if	SCONJ
ejpam-1857	176	2	one	one	PRON
ejpam-1857	176	3	wants	want	VERB
ejpam-1857	176	4	a	a	DET
ejpam-1857	176	5	point	point	NOUN
ejpam-1857	176	6	in	in	ADP
ejpam-1857	176	7	e(q(c	e(q(c	ADV
ejpam-1857	176	8	)	)	PUNCT
ejpam-1857	176	9	)	)	PUNCT
ejpam-1857	176	10	with	with	ADP
ejpam-1857	176	11	x	x	ADJ
ejpam-1857	176	12	-	-	NOUN
ejpam-1857	176	13	coordinate	coordinate	NOUN
ejpam-1857	176	14	equal	equal	ADJ
ejpam-1857	176	15	to	to	ADP
ejpam-1857	176	16	2(t2	2(t2	NUM
ejpam-1857	176	17	−	−	NOUN
ejpam-1857	176	18	1)2	1)2	NUM
ejpam-1857	176	19	,	,	PUNCT
ejpam-1857	176	20	one	one	PRON
ejpam-1857	176	21	needs	need	VERB
ejpam-1857	176	22	that	that	PRON
ejpam-1857	176	23	2(2(t2	2(2(t2	NOUN
ejpam-1857	176	24	−	−	PROPN
ejpam-1857	176	25	1)2	1)2	NUM
ejpam-1857	176	26	−	−	NOUN
ejpam-1857	176	27	4t2	4t2	NUM
ejpam-1857	176	28	)	)	PUNCT
ejpam-1857	177	1	=	=	SYM
ejpam-1857	178	1	4t4	4t4	ADJ
ejpam-1857	179	1	−	−	NOUN
ejpam-1857	179	2	16t2	16t2	NUM
ejpam-1857	180	1	+	+	CCONJ
ejpam-1857	180	2	4	4	NUM
ejpam-1857	180	3	,	,	PUNCT
ejpam-1857	180	4	is	be	AUX
ejpam-1857	180	5	a	a	DET
ejpam-1857	180	6	square	square	NOUN
ejpam-1857	180	7	,	,	PUNCT
ejpam-1857	180	8	say	say	VERB
ejpam-1857	180	9	=	=	SYM
ejpam-1857	180	10	s2	s2	PROPN
ejpam-1857	180	11	.	.	PUNCT
ejpam-1857	181	1	the	the	DET
ejpam-1857	181	2	equation	equation	NOUN
ejpam-1857	181	3	s2	s2	NOUN
ejpam-1857	181	4	=	=	SYM
ejpam-1857	181	5	4t4	4t4	NOUN
ejpam-1857	182	1	−	−	NOUN
ejpam-1857	182	2	16t2	16t2	NUM
ejpam-1857	183	1	+	+	CCONJ
ejpam-1857	183	2	4	4	NUM
ejpam-1857	183	3	defines	define	VERB
ejpam-1857	183	4	a	a	DET
ejpam-1857	183	5	curve	curve	NOUN
ejpam-1857	183	6	c1	c1	NOUN
ejpam-1857	183	7	of	of	ADP
ejpam-1857	183	8	genus	genus	PROPN
ejpam-1857	183	9	1	1	NUM
ejpam-1857	183	10	with	with	ADP
ejpam-1857	183	11	infinitely	infinitely	ADV
ejpam-1857	183	12	many	many	ADJ
ejpam-1857	183	13	rational	rational	ADJ
ejpam-1857	183	14	points	point	NOUN
ejpam-1857	183	15	,	,	PUNCT
ejpam-1857	183	16	namely	namely	ADV
ejpam-1857	183	17	this	this	PRON
ejpam-1857	183	18	is	be	AUX
ejpam-1857	183	19	the	the	DET
ejpam-1857	183	20	example	example	NOUN
ejpam-1857	183	21	discussed	discuss	VERB
ejpam-1857	183	22	in	in	ADP
ejpam-1857	183	23	section	section	NOUN
ejpam-1857	183	24	3	3	NUM
ejpam-1857	183	25	.	.	PUNCT
ejpam-1857	184	1	the	the	DET
ejpam-1857	184	2	two	two	NUM
ejpam-1857	184	3	points	point	NOUN
ejpam-1857	184	4	s1	s1	NOUN
ejpam-1857	184	5	(	(	PUNCT
ejpam-1857	184	6	defined	define	VERB
ejpam-1857	184	7	earlier	early	ADV
ejpam-1857	184	8	)	)	PUNCT
ejpam-1857	184	9	and	and	CCONJ
ejpam-1857	184	10	s3	s3	PROPN
ejpam-1857	184	11	:	:	PUNCT
ejpam-1857	184	12	=	=	X
ejpam-1857	184	13	(	(	PUNCT
ejpam-1857	184	14	2(t2	2(t2	NUM
ejpam-1857	184	15	−	−	NOUN
ejpam-1857	184	16	1)2	1)2	NUM
ejpam-1857	184	17	,	,	PUNCT
ejpam-1857	184	18	(	(	PUNCT
ejpam-1857	184	19	t2	t2	NOUN
ejpam-1857	184	20	−	−	PROPN
ejpam-1857	184	21	1)2s	1)2s	NUM
ejpam-1857	184	22	)	)	PUNCT
ejpam-1857	184	23	are	be	AUX
ejpam-1857	184	24	independent	independent	ADJ
ejpam-1857	184	25	in	in	ADP
ejpam-1857	184	26	e(q(c1	e(q(c1	NOUN
ejpam-1857	184	27	)	)	PUNCT
ejpam-1857	184	28	)	)	PUNCT
ejpam-1857	184	29	,	,	PUNCT
ejpam-1857	184	30	since	since	SCONJ
ejpam-1857	184	31	their	their	PRON
ejpam-1857	184	32	images	image	NOUN
ejpam-1857	184	33	after	after	ADP
ejpam-1857	184	34	a	a	DET
ejpam-1857	184	35	given	give	VERB
ejpam-1857	184	36	specialization	specialization	NOUN
ejpam-1857	184	37	are	be	AUX
ejpam-1857	184	38	independent	independent	ADJ
ejpam-1857	184	39	.	.	PUNCT
ejpam-1857	185	1	instead	instead	ADV
ejpam-1857	185	2	of	of	ADP
ejpam-1857	185	3	a	a	DET
ejpam-1857	185	4	point	point	NOUN
ejpam-1857	185	5	with	with	ADP
ejpam-1857	185	6	x	x	ADJ
ejpam-1857	185	7	-	-	NOUN
ejpam-1857	185	8	coordinate	coordinate	NOUN
ejpam-1857	185	9	equal	equal	ADJ
ejpam-1857	185	10	to	to	ADP
ejpam-1857	185	11	2(t2−1)2	2(t2−1)2	NUM
ejpam-1857	185	12	,	,	PUNCT
ejpam-1857	185	13	other	other	ADJ
ejpam-1857	185	14	slightly	slightly	ADV
ejpam-1857	185	15	simpler	simple	ADJ
ejpam-1857	185	16	points	point	NOUN
ejpam-1857	185	17	work	work	NOUN
ejpam-1857	185	18	as	as	ADV
ejpam-1857	185	19	well	well	ADV
ejpam-1857	185	20	.	.	PUNCT
ejpam-1857	186	1	for	for	ADP
ejpam-1857	186	2	example	example	NOUN
ejpam-1857	186	3	,	,	PUNCT
ejpam-1857	186	4	take	take	VERB
ejpam-1857	186	5	x	x	X
ejpam-1857	186	6	=	=	SYM
ejpam-1857	186	7	8t2	8t2	NUM
ejpam-1857	186	8	.	.	PUNCT
ejpam-1857	187	1	then	then	ADV
ejpam-1857	187	2	one	one	PRON
ejpam-1857	187	3	wants	want	VERB
ejpam-1857	187	4	2(8t2	2(8t2	PRON
ejpam-1857	187	5	−	−	PROPN
ejpam-1857	188	1	(	(	PUNCT
ejpam-1857	188	2	t2	t2	NOUN
ejpam-1857	188	3	−	−	PROPN
ejpam-1857	188	4	1)2	1)2	NUM
ejpam-1857	188	5	)	)	PUNCT
ejpam-1857	188	6	to	to	PART
ejpam-1857	188	7	be	be	AUX
ejpam-1857	188	8	a	a	DET
ejpam-1857	188	9	square	square	NOUN
ejpam-1857	188	10	.	.	PUNCT
ejpam-1857	189	1	the	the	DET
ejpam-1857	189	2	curve	curve	NOUN
ejpam-1857	189	3	c2	c2	PROPN
ejpam-1857	189	4	defined	define	VERB
ejpam-1857	189	5	by	by	ADP
ejpam-1857	189	6	u2	u2	PROPN
ejpam-1857	189	7	=	=	SYM
ejpam-1857	189	8	−2t4	−2t4	PROPN
ejpam-1857	189	9	+	+	NOUN
ejpam-1857	189	10	20t2	20t2	NUM
ejpam-1857	189	11	−	−	NOUN
ejpam-1857	189	12	2	2	NUM
ejpam-1857	189	13	has	have	VERB
ejpam-1857	189	14	infinitely	infinitely	ADV
ejpam-1857	189	15	many	many	ADJ
ejpam-1857	189	16	rational	rational	ADJ
ejpam-1857	189	17	many	many	ADJ
ejpam-1857	189	18	points	point	NOUN
ejpam-1857	189	19	as	as	ADV
ejpam-1857	189	20	well	well	ADV
ejpam-1857	189	21	.	.	PUNCT
ejpam-1857	190	1	it	it	PRON
ejpam-1857	190	2	corresponds	correspond	VERB
ejpam-1857	190	3	to	to	ADP
ejpam-1857	190	4	the	the	DET
ejpam-1857	190	5	elliptic	elliptic	ADJ
ejpam-1857	190	6	curve	curve	NOUN
ejpam-1857	190	7	given	give	VERB
ejpam-1857	190	8	by	by	ADP
ejpam-1857	190	9	y2	y2	PROPN
ejpam-1857	190	10	=	=	SYM
ejpam-1857	190	11	x(x	x(x	PROPN
ejpam-1857	191	1	+	+	CCONJ
ejpam-1857	191	2	3	3	NUM
ejpam-1857	191	3	2	2	NUM
ejpam-1857	191	4	)	)	PUNCT
ejpam-1857	191	5	(	(	PUNCT
ejpam-1857	191	6	x	x	X
ejpam-1857	192	1	+	+	NOUN
ejpam-1857	192	2	1	1	NUM
ejpam-1857	192	3	2	2	NUM
ejpam-1857	192	4	)	)	PUNCT
ejpam-1857	192	5	,	,	PUNCT
ejpam-1857	192	6	which	which	PRON
ejpam-1857	192	7	has	have	VERB
ejpam-1857	192	8	z×	z×	NUM
ejpam-1857	192	9	(	(	PUNCT
ejpam-1857	192	10	z/2z)×	z/2z)×	X
ejpam-1857	192	11	(	(	PUNCT
ejpam-1857	192	12	z/2z	z/2z	NUM
ejpam-1857	192	13	)	)	PUNCT
ejpam-1857	192	14	,	,	PUNCT
ejpam-1857	192	15	as	as	ADP
ejpam-1857	192	16	its	its	PRON
ejpam-1857	192	17	group	group	NOUN
ejpam-1857	192	18	of	of	ADP
ejpam-1857	192	19	rational	rational	ADJ
ejpam-1857	192	20	points	point	NOUN
ejpam-1857	192	21	.	.	PUNCT
ejpam-1857	193	1	in	in	ADP
ejpam-1857	193	2	e(q(c2	e(q(c2	NOUN
ejpam-1857	193	3	)	)	PUNCT
ejpam-1857	193	4	)	)	PUNCT
ejpam-1857	193	5	one	one	PRON
ejpam-1857	193	6	has	have	VERB
ejpam-1857	193	7	by	by	ADP
ejpam-1857	193	8	construction	construction	NOUN
ejpam-1857	193	9	the	the	DET
ejpam-1857	193	10	point	point	NOUN
ejpam-1857	193	11	s4	s4	NOUN
ejpam-1857	193	12	:	:	PUNCT
ejpam-1857	193	13	=	=	SYM
ejpam-1857	193	14	(	(	PUNCT
ejpam-1857	193	15	8t2	8t2	NUM
ejpam-1857	193	16	,	,	PUNCT
ejpam-1857	193	17	4t2s	4t2s	NOUN
ejpam-1857	193	18	)	)	PUNCT
ejpam-1857	193	19	which	which	PRON
ejpam-1857	193	20	turns	turn	VERB
ejpam-1857	193	21	out	out	ADP
ejpam-1857	193	22	to	to	PART
ejpam-1857	193	23	be	be	AUX
ejpam-1857	193	24	independent	independent	ADJ
ejpam-1857	193	25	of	of	ADP
ejpam-1857	193	26	s1	s1	NOUN
ejpam-1857	193	27	.	.	PUNCT
ejpam-1857	194	1	so	so	ADV
ejpam-1857	194	2	again	again	ADV
ejpam-1857	194	3	,	,	PUNCT
ejpam-1857	194	4	specialization	specialization	NOUN
ejpam-1857	194	5	yields	yield	NOUN
ejpam-1857	194	6	infinitely	infinitely	ADV
ejpam-1857	194	7	many	many	ADJ
ejpam-1857	194	8	example	example	NOUN
ejpam-1857	194	9	of	of	ADP
ejpam-1857	194	10	rank	rank	NOUN
ejpam-1857	194	11	at	at	ADV
ejpam-1857	194	12	least	least	ADV
ejpam-1857	194	13	2	2	NUM
ejpam-1857	194	14	over	over	ADP
ejpam-1857	194	15	q.	q.	NOUN
ejpam-1857	194	16	by	by	ADP
ejpam-1857	194	17	taking	take	VERB
ejpam-1857	194	18	x	x	PUNCT
ejpam-1857	194	19	=	=	PUNCT
ejpam-1857	194	20	t2	t2	NOUN
ejpam-1857	194	21	−	−	PROPN
ejpam-1857	194	22	1	1	NUM
ejpam-1857	194	23	,	,	PUNCT
ejpam-1857	194	24	one	one	PRON
ejpam-1857	194	25	needs	need	VERB
ejpam-1857	194	26	that	that	SCONJ
ejpam-1857	194	27	3t4	3t4	NUM
ejpam-1857	194	28	−	−	NOUN
ejpam-1857	194	29	5t2	5t2	NUM
ejpam-1857	195	1	−	−	NOUN
ejpam-1857	195	2	2	2	NUM
ejpam-1857	195	3	is	be	AUX
ejpam-1857	195	4	a	a	DET
ejpam-1857	195	5	square	square	NOUN
ejpam-1857	195	6	,	,	PUNCT
ejpam-1857	195	7	say	say	VERB
ejpam-1857	195	8	=	=	SYM
ejpam-1857	195	9	s2	s2	PROPN
ejpam-1857	195	10	.	.	PUNCT
ejpam-1857	196	1	the	the	DET
ejpam-1857	196	2	curve	curve	NOUN
ejpam-1857	196	3	c3	c3	PROPN
ejpam-1857	196	4	defined	define	VERB
ejpam-1857	196	5	by	by	ADP
ejpam-1857	196	6	s2	s2	NOUN
ejpam-1857	196	7	=	=	VERB
ejpam-1857	196	8	3t4−5t2−2	3t4−5t2−2	NUM
ejpam-1857	196	9	has	have	VERB
ejpam-1857	196	10	infinitely	infinitely	ADV
ejpam-1857	196	11	many	many	ADJ
ejpam-1857	196	12	rational	rational	ADJ
ejpam-1857	196	13	points	point	NOUN
ejpam-1857	196	14	as	as	ADV
ejpam-1857	196	15	well	well	ADV
ejpam-1857	196	16	.	.	PUNCT
ejpam-1857	197	1	it	it	PRON
ejpam-1857	197	2	corresponds	correspond	VERB
ejpam-1857	197	3	to	to	ADP
ejpam-1857	197	4	the	the	DET
ejpam-1857	197	5	elliptic	elliptic	ADJ
ejpam-1857	197	6	curve	curve	NOUN
ejpam-1857	197	7	given	give	VERB
ejpam-1857	197	8	by	by	ADP
ejpam-1857	197	9	y2	y2	PROPN
ejpam-1857	197	10	=	=	SYM
ejpam-1857	198	1	(	(	PUNCT
ejpam-1857	198	2	x	x	X
ejpam-1857	198	3	−	−	NOUN
ejpam-1857	198	4	3)(x2	3)(x2	NUM
ejpam-1857	199	1	+	+	CCONJ
ejpam-1857	199	2	4x	4x	NUM
ejpam-1857	199	3	+	+	NOUN
ejpam-1857	199	4	28	28	NUM
ejpam-1857	199	5	)	)	PUNCT
ejpam-1857	199	6	,	,	PUNCT
ejpam-1857	199	7	which	which	PRON
ejpam-1857	199	8	has	have	VERB
ejpam-1857	199	9	z×	z×	NUM
ejpam-1857	199	10	(	(	PUNCT
ejpam-1857	199	11	z/2z	z/2z	NUM
ejpam-1857	199	12	)	)	PUNCT
ejpam-1857	199	13	,	,	PUNCT
ejpam-1857	199	14	f.	f.	PROPN
ejpam-1857	199	15	izadi	izadi	PROPN
ejpam-1857	199	16	,	,	PUNCT
ejpam-1857	199	17	k.	k.	PROPN
ejpam-1857	199	18	nabardi	nabardi	PROPN
ejpam-1857	199	19	/	/	SYM
ejpam-1857	199	20	eur	eur	PROPN
ejpam-1857	199	21	.	.	PUNCT
ejpam-1857	200	1	j.	j.	PROPN
ejpam-1857	200	2	pure	pure	PROPN
ejpam-1857	200	3	appl	appl	PROPN
ejpam-1857	200	4	.	.	PROPN
ejpam-1857	200	5	math	math	PROPN
ejpam-1857	200	6	,	,	PUNCT
ejpam-1857	200	7	7	7	NUM
ejpam-1857	200	8	(	(	PUNCT
ejpam-1857	200	9	2014	2014	NUM
ejpam-1857	200	10	)	)	PUNCT
ejpam-1857	200	11	,	,	PUNCT
ejpam-1857	200	12	131	131	NUM
ejpam-1857	200	13	-	-	SYM
ejpam-1857	200	14	139	139	NUM
ejpam-1857	200	15	138	138	NUM
ejpam-1857	200	16	as	as	ADP
ejpam-1857	200	17	its	its	PRON
ejpam-1857	200	18	group	group	NOUN
ejpam-1857	200	19	or	or	CCONJ
ejpam-1857	200	20	rational	rational	ADJ
ejpam-1857	200	21	points	point	NOUN
ejpam-1857	200	22	.	.	PUNCT
ejpam-1857	201	1	so	so	ADV
ejpam-1857	201	2	in	in	ADP
ejpam-1857	201	3	e(q(c3	e(q(c3	ADJ
ejpam-1857	201	4	)	)	PUNCT
ejpam-1857	201	5	)	)	PUNCT
ejpam-1857	201	6	one	one	PRON
ejpam-1857	201	7	has	have	VERB
ejpam-1857	201	8	by	by	ADP
ejpam-1857	201	9	construction	construction	NOUN
ejpam-1857	201	10	the	the	DET
ejpam-1857	201	11	point	point	NOUN
ejpam-1857	201	12	s5	s5	PROPN
ejpam-1857	201	13	:	:	PUNCT
ejpam-1857	201	14	=	=	SYM
ejpam-1857	201	15	(	(	PUNCT
ejpam-1857	201	16	t2	t2	NOUN
ejpam-1857	201	17	−	−	PROPN
ejpam-1857	201	18	1	1	NUM
ejpam-1857	201	19	,	,	PUNCT
ejpam-1857	201	20	(	(	PUNCT
ejpam-1857	201	21	t2	t2	NOUN
ejpam-1857	201	22	−	−	PROPN
ejpam-1857	201	23	1)2s	1)2s	NUM
ejpam-1857	201	24	)	)	PUNCT
ejpam-1857	201	25	which	which	PRON
ejpam-1857	201	26	turns	turn	VERB
ejpam-1857	201	27	out	out	ADP
ejpam-1857	201	28	to	to	PART
ejpam-1857	201	29	be	be	AUX
ejpam-1857	201	30	independent	independent	ADJ
ejpam-1857	201	31	of	of	ADP
ejpam-1857	201	32	s1	s1	NOUN
ejpam-1857	201	33	.	.	PUNCT
ejpam-1857	202	1	now	now	ADV
ejpam-1857	202	2	we	we	PRON
ejpam-1857	202	3	would	would	AUX
ejpam-1857	202	4	like	like	VERB
ejpam-1857	202	5	to	to	PART
ejpam-1857	202	6	follow	follow	VERB
ejpam-1857	202	7	the	the	DET
ejpam-1857	202	8	method	method	NOUN
ejpam-1857	202	9	described	describe	VERB
ejpam-1857	202	10	on	on	ADP
ejpam-1857	202	11	[	[	X
ejpam-1857	202	12	8	8	NUM
ejpam-1857	202	13	,	,	PUNCT
ejpam-1857	202	14	page	page	NOUN
ejpam-1857	202	15	89	89	NUM
ejpam-1857	202	16	]	]	PUNCT
ejpam-1857	202	17	,	,	PUNCT
ejpam-1857	202	18	to	to	PART
ejpam-1857	202	19	find	find	VERB
ejpam-1857	202	20	a	a	DET
ejpam-1857	202	21	subfamily	subfamily	NOUN
ejpam-1857	202	22	of	of	ADP
ejpam-1857	202	23	rank	rank	NOUN
ejpam-1857	202	24	at	at	ADV
ejpam-1857	202	25	least	least	ADV
ejpam-1857	202	26	2	2	NUM
ejpam-1857	202	27	.	.	X
ejpam-1857	202	28	consider	consider	VERB
ejpam-1857	202	29	a	a	PRON
ejpam-1857	202	30	=	=	NOUN
ejpam-1857	202	31	−(t2	−(t2	PUNCT
ejpam-1857	203	1	+	+	NOUN
ejpam-1857	203	2	1)2	1)2	NUM
ejpam-1857	203	3	and	and	CCONJ
ejpam-1857	203	4	b	b	X
ejpam-1857	203	5	=	=	SYM
ejpam-1857	203	6	4t2(t2	4t2(t2	NUM
ejpam-1857	204	1	−	−	NOUN
ejpam-1857	204	2	1)2	1)2	NUM
ejpam-1857	204	3	.	.	PUNCT
ejpam-1857	205	1	let	let	VERB
ejpam-1857	205	2	b	b	NOUN
ejpam-1857	205	3	=	=	SYM
ejpam-1857	205	4	b1	b1	NOUN
ejpam-1857	205	5	b2	b2	NOUN
ejpam-1857	205	6	,	,	PUNCT
ejpam-1857	205	7	if	if	SCONJ
ejpam-1857	205	8	one	one	PRON
ejpam-1857	205	9	can	can	AUX
ejpam-1857	205	10	find	find	VERB
ejpam-1857	205	11	integers	integer	NOUN
ejpam-1857	205	12	m	m	PRON
ejpam-1857	205	13	,	,	PUNCT
ejpam-1857	205	14	n	n	CCONJ
ejpam-1857	205	15	,	,	PUNCT
ejpam-1857	205	16	e	e	X
ejpam-1857	205	17	such	such	ADJ
ejpam-1857	205	18	that	that	DET
ejpam-1857	205	19	gcd(m	gcd(m	NOUN
ejpam-1857	205	20	,	,	PUNCT
ejpam-1857	205	21	e	e	NOUN
ejpam-1857	205	22	)	)	PUNCT
ejpam-1857	205	23	=	=	SYM
ejpam-1857	205	24	gcd(n	gcd(n	NOUN
ejpam-1857	205	25	,	,	PUNCT
ejpam-1857	205	26	e	e	NOUN
ejpam-1857	205	27	)	)	PUNCT
ejpam-1857	205	28	=	=	SYM
ejpam-1857	205	29	gcd(b1	gcd(b1	X
ejpam-1857	205	30	,	,	PUNCT
ejpam-1857	205	31	e	e	NOUN
ejpam-1857	205	32	)	)	PUNCT
ejpam-1857	205	33	=	=	SYM
ejpam-1857	205	34	1	1	NUM
ejpam-1857	205	35	and	and	CCONJ
ejpam-1857	205	36	b1m4	b1m4	PROPN
ejpam-1857	206	1	+	+	CCONJ
ejpam-1857	206	2	am2e2	am2e2	PROPN
ejpam-1857	206	3	+	+	CCONJ
ejpam-1857	206	4	b2e4	b2e4	NOUN
ejpam-1857	206	5	=	=	SYM
ejpam-1857	206	6	n2	n2	NOUN
ejpam-1857	206	7	,	,	PUNCT
ejpam-1857	206	8	then	then	ADV
ejpam-1857	206	9	(	(	PUNCT
ejpam-1857	206	10	b1m2	b1m2	X
ejpam-1857	206	11	/	/	SYM
ejpam-1857	206	12	e2	e2	PROPN
ejpam-1857	206	13	,	,	PUNCT
ejpam-1857	206	14	b1mn	b1mn	X
ejpam-1857	206	15	/	/	SYM
ejpam-1857	206	16	e3	e3	NOUN
ejpam-1857	206	17	)	)	PUNCT
ejpam-1857	206	18	is	be	AUX
ejpam-1857	206	19	a	a	DET
ejpam-1857	206	20	point	point	NOUN
ejpam-1857	206	21	on	on	ADP
ejpam-1857	206	22	(	(	PUNCT
ejpam-1857	206	23	9	9	NUM
ejpam-1857	206	24	)	)	PUNCT
ejpam-1857	206	25	as	as	ADV
ejpam-1857	206	26	well	well	ADV
ejpam-1857	206	27	.	.	PUNCT
ejpam-1857	207	1	we	we	PRON
ejpam-1857	207	2	let	let	VERB
ejpam-1857	207	3	b1	b1	NOUN
ejpam-1857	207	4	=	=	SYM
ejpam-1857	207	5	(	(	PUNCT
ejpam-1857	207	6	t2	t2	NOUN
ejpam-1857	207	7	−	−	PROPN
ejpam-1857	207	8	1	1	NUM
ejpam-1857	207	9	)	)	PUNCT
ejpam-1857	207	10	,	,	PUNCT
ejpam-1857	207	11	m	m	VERB
ejpam-1857	207	12	=	=	SYM
ejpam-1857	207	13	2	2	NUM
ejpam-1857	207	14	and	and	CCONJ
ejpam-1857	207	15	e	e	NOUN
ejpam-1857	207	16	=	=	NOUN
ejpam-1857	207	17	1	1	X
ejpam-1857	207	18	.	.	PUNCT
ejpam-1857	208	1	after	after	ADP
ejpam-1857	208	2	a	a	DET
ejpam-1857	208	3	little	little	ADJ
ejpam-1857	208	4	computation	computation	NOUN
ejpam-1857	208	5	one	one	NOUN
ejpam-1857	208	6	gets	get	VERB
ejpam-1857	208	7	4t2	4t2	NUM
ejpam-1857	208	8	−	−	NOUN
ejpam-1857	208	9	20=	20=	NUM
ejpam-1857	208	10	n2	n2	NOUN
ejpam-1857	208	11	.	.	PUNCT
ejpam-1857	209	1	(	(	PUNCT
ejpam-1857	209	2	10	10	NUM
ejpam-1857	209	3	)	)	PUNCT
ejpam-1857	209	4	a	a	DET
ejpam-1857	209	5	particular	particular	ADJ
ejpam-1857	209	6	solution	solution	NOUN
ejpam-1857	209	7	for	for	ADP
ejpam-1857	209	8	(	(	PUNCT
ejpam-1857	209	9	10	10	NUM
ejpam-1857	209	10	)	)	PUNCT
ejpam-1857	209	11	is	be	AUX
ejpam-1857	209	12	(	(	PUNCT
ejpam-1857	209	13	t	t	PROPN
ejpam-1857	209	14	,	,	PUNCT
ejpam-1857	209	15	n	n	CCONJ
ejpam-1857	209	16	)	)	PUNCT
ejpam-1857	209	17	=	=	SYM
ejpam-1857	209	18	(	(	PUNCT
ejpam-1857	209	19	3	3	NUM
ejpam-1857	209	20	,	,	PUNCT
ejpam-1857	209	21	4	4	NUM
ejpam-1857	209	22	)	)	PUNCT
ejpam-1857	209	23	.	.	PUNCT
ejpam-1857	210	1	using	use	VERB
ejpam-1857	210	2	this	this	DET
ejpam-1857	210	3	solution	solution	NOUN
ejpam-1857	210	4	we	we	PRON
ejpam-1857	210	5	can	can	AUX
ejpam-1857	210	6	parameterize	parameterize	VERB
ejpam-1857	210	7	the	the	DET
ejpam-1857	210	8	corresponding	correspond	VERB
ejpam-1857	210	9	hyperbola	hyperbola	PROPN
ejpam-1857	210	10	as	as	ADP
ejpam-1857	210	11	following	follow	VERB
ejpam-1857	210	12	:	:	PUNCT
ejpam-1857	211	1	t	t	PROPN
ejpam-1857	211	2	=	=	PUNCT
ejpam-1857	211	3	3m2	3m2	NUM
ejpam-1857	211	4	−	−	NOUN
ejpam-1857	211	5	8m+	8m+	NUM
ejpam-1857	211	6	12	12	NUM
ejpam-1857	211	7	m2	m2	PROPN
ejpam-1857	211	8	−	−	PROPN
ejpam-1857	211	9	4	4	NUM
ejpam-1857	211	10	,	,	PUNCT
ejpam-1857	211	11	n	n	NOUN
ejpam-1857	211	12	=	=	SYM
ejpam-1857	211	13	4(m2	4(m2	NUM
ejpam-1857	211	14	−	−	NOUN
ejpam-1857	211	15	6m+	6m+	NUM
ejpam-1857	211	16	4	4	NUM
ejpam-1857	211	17	)	)	PUNCT
ejpam-1857	211	18	4−m2	4−m2	NUM
ejpam-1857	211	19	,	,	PUNCT
ejpam-1857	211	20	m	m	VERB
ejpam-1857	211	21	∈q	∈q	X
ejpam-1857	211	22	.	.	PUNCT
ejpam-1857	212	1	(	(	PUNCT
ejpam-1857	212	2	11	11	NUM
ejpam-1857	212	3	)	)	PUNCT
ejpam-1857	212	4	it	it	PRON
ejpam-1857	212	5	is	be	AUX
ejpam-1857	212	6	clear	clear	ADJ
ejpam-1857	212	7	that	that	SCONJ
ejpam-1857	212	8	|t|	|t|	VERB
ejpam-1857	212	9	≥	≥	NOUN
ejpam-1857	212	10	p	p	NOUN
ejpam-1857	212	11	5	5	NUM
ejpam-1857	212	12	or	or	CCONJ
ejpam-1857	212	13	equivalency	equivalency	NOUN
ejpam-1857	212	14	|m|	|m|	VERB
ejpam-1857	212	15	>	>	X
ejpam-1857	212	16	2	2	NUM
ejpam-1857	212	17	.	.	PUNCT
ejpam-1857	212	18	by	by	ADP
ejpam-1857	212	19	taking	take	VERB
ejpam-1857	212	20	any	any	DET
ejpam-1857	212	21	rational	rational	ADJ
ejpam-1857	212	22	values	value	NOUN
ejpam-1857	212	23	m	m	VERB
ejpam-1857	212	24	>	>	X
ejpam-1857	212	25	2	2	NUM
ejpam-1857	212	26	,	,	PUNCT
ejpam-1857	212	27	we	we	PRON
ejpam-1857	212	28	get	get	VERB
ejpam-1857	212	29	a	a	DET
ejpam-1857	212	30	rational	rational	ADJ
ejpam-1857	212	31	value	value	NOUN
ejpam-1857	212	32	of	of	ADP
ejpam-1857	212	33	t	t	PROPN
ejpam-1857	212	34	as	as	ADP
ejpam-1857	212	35	(	(	PUNCT
ejpam-1857	212	36	11	11	NUM
ejpam-1857	212	37	)	)	PUNCT
ejpam-1857	212	38	,	,	PUNCT
ejpam-1857	212	39	we	we	PRON
ejpam-1857	212	40	see	see	VERB
ejpam-1857	212	41	that	that	SCONJ
ejpam-1857	212	42	the	the	DET
ejpam-1857	212	43	point	point	NOUN
ejpam-1857	212	44	(	(	PUNCT
ejpam-1857	212	45	4(t2	4(t2	NOUN
ejpam-1857	212	46	−	−	NOUN
ejpam-1857	212	47	1	1	NUM
ejpam-1857	212	48	)	)	PUNCT
ejpam-1857	212	49	,	,	PUNCT
ejpam-1857	212	50	2(t2	2(t2	NUM
ejpam-1857	212	51	−	−	PROPN
ejpam-1857	212	52	1)n	1)n	NUM
ejpam-1857	212	53	)	)	PUNCT
ejpam-1857	212	54	is	be	AUX
ejpam-1857	212	55	on	on	ADP
ejpam-1857	212	56	(	(	PUNCT
ejpam-1857	212	57	9	9	NUM
ejpam-1857	212	58	)	)	PUNCT
ejpam-1857	212	59	.	.	PUNCT
ejpam-1857	213	1	in	in	ADP
ejpam-1857	213	2	order	order	NOUN
ejpam-1857	213	3	to	to	PART
ejpam-1857	213	4	show	show	VERB
ejpam-1857	213	5	that	that	SCONJ
ejpam-1857	213	6	the	the	DET
ejpam-1857	213	7	points	point	NOUN
ejpam-1857	213	8	s1	s1	NOUN
ejpam-1857	213	9	=	=	SYM
ejpam-1857	213	10	(	(	PUNCT
ejpam-1857	213	11	(	(	PUNCT
ejpam-1857	213	12	t2	t2	NOUN
ejpam-1857	213	13	+	+	CCONJ
ejpam-1857	213	14	1)2	1)2	NUM
ejpam-1857	213	15	,	,	PUNCT
ejpam-1857	213	16	2t(t4	2t(t4	NUM
ejpam-1857	213	17	−	−	NOUN
ejpam-1857	213	18	1	1	NUM
ejpam-1857	213	19	)	)	PUNCT
ejpam-1857	213	20	)	)	PUNCT
ejpam-1857	213	21	and	and	CCONJ
ejpam-1857	213	22	s6	s6	PROPN
ejpam-1857	213	23	:	:	PUNCT
ejpam-1857	213	24	=	=	SYM
ejpam-1857	213	25	(	(	PUNCT
ejpam-1857	213	26	4(t2	4(t2	NOUN
ejpam-1857	213	27	−	−	NOUN
ejpam-1857	213	28	1	1	NUM
ejpam-1857	213	29	)	)	PUNCT
ejpam-1857	213	30	,	,	PUNCT
ejpam-1857	213	31	2(t2	2(t2	NUM
ejpam-1857	213	32	−	−	PROPN
ejpam-1857	213	33	1)n	1)n	NUM
ejpam-1857	213	34	)	)	PUNCT
ejpam-1857	213	35	are	be	AUX
ejpam-1857	213	36	independent	independent	ADJ
ejpam-1857	213	37	we	we	PRON
ejpam-1857	213	38	use	use	VERB
ejpam-1857	213	39	specialization	specialization	NOUN
ejpam-1857	213	40	(	(	PUNCT
ejpam-1857	213	41	m	m	PROPN
ejpam-1857	213	42	,	,	PUNCT
ejpam-1857	213	43	t	t	PROPN
ejpam-1857	213	44	,	,	PUNCT
ejpam-1857	213	45	n	n	CCONJ
ejpam-1857	213	46	)	)	PUNCT
ejpam-1857	213	47	=	=	SYM
ejpam-1857	213	48	(	(	PUNCT
ejpam-1857	213	49	4	4	NUM
ejpam-1857	213	50	,	,	PUNCT
ejpam-1857	213	51	7/3,4/3	7/3,4/3	NUM
ejpam-1857	213	52	)	)	PUNCT
ejpam-1857	213	53	.	.	PUNCT
ejpam-1857	214	1	this	this	PRON
ejpam-1857	214	2	gives	give	VERB
ejpam-1857	214	3	rise	rise	NOUN
ejpam-1857	214	4	to	to	ADP
ejpam-1857	214	5	following	follow	VERB
ejpam-1857	214	6	elliptic	elliptic	ADJ
ejpam-1857	214	7	curve	curve	NOUN
ejpam-1857	214	8	e7/3	e7/3	NOUN
ejpam-1857	214	9	:	:	PUNCT
ejpam-1857	214	10	y2	y2	X
ejpam-1857	215	1	=	=	SYM
ejpam-1857	216	1	x3	x3	ADJ
ejpam-1857	216	2	−	−	PROPN
ejpam-1857	217	1	3364	3364	NUM
ejpam-1857	217	2	81	81	NUM
ejpam-1857	217	3	x2	x2	NOUN
ejpam-1857	218	1	+	+	CCONJ
ejpam-1857	218	2	313600	313600	NUM
ejpam-1857	218	3	729	729	NUM
ejpam-1857	218	4	x	x	NOUN
ejpam-1857	218	5	,	,	PUNCT
ejpam-1857	218	6	and	and	CCONJ
ejpam-1857	218	7	the	the	DET
ejpam-1857	218	8	points	point	NOUN
ejpam-1857	218	9	p	p	X
ejpam-1857	218	10	=	=	X
ejpam-1857	218	11	(	(	PUNCT
ejpam-1857	218	12	3364	3364	NUM
ejpam-1857	218	13	81	81	NUM
ejpam-1857	218	14	,	,	PUNCT
ejpam-1857	218	15	32480	32480	NUM
ejpam-1857	218	16	243	243	NUM
ejpam-1857	218	17	)	)	PUNCT
ejpam-1857	218	18	,	,	PUNCT
ejpam-1857	218	19	q	q	NOUN
ejpam-1857	218	20	=	=	PUNCT
ejpam-1857	218	21	(	(	PUNCT
ejpam-1857	218	22	160	160	NUM
ejpam-1857	218	23	9	9	NUM
ejpam-1857	218	24	,	,	PUNCT
ejpam-1857	218	25	320	320	NUM
ejpam-1857	218	26	27	27	NUM
ejpam-1857	218	27	)	)	PUNCT
ejpam-1857	218	28	.	.	PUNCT
ejpam-1857	219	1	now	now	ADV
ejpam-1857	219	2	the	the	DET
ejpam-1857	219	3	associated	associated	ADJ
ejpam-1857	219	4	height	height	NOUN
ejpam-1857	219	5	matrix	matrix	NOUN
ejpam-1857	219	6	of	of	ADP
ejpam-1857	219	7	the	the	DET
ejpam-1857	219	8	above	above	ADJ
ejpam-1857	219	9	points	point	NOUN
ejpam-1857	219	10	has	have	VERB
ejpam-1857	219	11	non	non	ADJ
ejpam-1857	219	12	-	-	ADJ
ejpam-1857	219	13	zero	zero	NUM
ejpam-1857	219	14	determinant	determinant	ADJ
ejpam-1857	219	15	1.011058	1.011058	NUM
ejpam-1857	219	16	showing	show	VERB
ejpam-1857	219	17	that	that	SCONJ
ejpam-1857	219	18	they	they	PRON
ejpam-1857	219	19	are	be	AUX
ejpam-1857	219	20	independent	independent	ADJ
ejpam-1857	219	21	.	.	PUNCT
ejpam-1857	220	1	by	by	ADP
ejpam-1857	220	2	letting	let	VERB
ejpam-1857	220	3	m	m	PRON
ejpam-1857	220	4	as	as	ADP
ejpam-1857	220	5	m	m	PROPN
ejpam-1857	220	6	=	=	SYM
ejpam-1857	220	7	α	α	PROPN
ejpam-1857	220	8	/	/	SYM
ejpam-1857	220	9	β	β	NOUN
ejpam-1857	220	10	with	with	ADP
ejpam-1857	220	11	α	α	PROPN
ejpam-1857	220	12	,	,	PUNCT
ejpam-1857	220	13	β	β	X
ejpam-1857	220	14	≤	≤	NUM
ejpam-1857	220	15	350	350	NUM
ejpam-1857	220	16	,	,	PUNCT
ejpam-1857	220	17	we	we	PRON
ejpam-1857	220	18	found	find	VERB
ejpam-1857	220	19	4	4	NUM
ejpam-1857	220	20	curves	curve	NOUN
ejpam-1857	220	21	of	of	ADP
ejpam-1857	220	22	rank	rank	NOUN
ejpam-1857	220	23	6	6	NUM
ejpam-1857	220	24	.	.	PUNCT
ejpam-1857	221	1	in	in	ADP
ejpam-1857	221	2	particular	particular	ADJ
ejpam-1857	221	3	m	m	PROPN
ejpam-1857	221	4	=	=	SYM
ejpam-1857	221	5	128/33	128/33	NUM
ejpam-1857	221	6	,	,	PUNCT
ejpam-1857	221	7	m	m	VERB
ejpam-1857	221	8	=	=	SYM
ejpam-1857	221	9	152/27	152/27	NUM
ejpam-1857	221	10	,	,	PUNCT
ejpam-1857	221	11	m	m	VERB
ejpam-1857	221	12	=	=	NOUN
ejpam-1857	221	13	252/29	252/29	NUM
ejpam-1857	221	14	,	,	PUNCT
ejpam-1857	221	15	and	and	CCONJ
ejpam-1857	221	16	m	m	PROPN
ejpam-1857	221	17	=	=	NOUN
ejpam-1857	222	1	348/71	348/71	NUM
ejpam-1857	222	2	we	we	PRON
ejpam-1857	222	3	get	get	VERB
ejpam-1857	222	4	the	the	DET
ejpam-1857	222	5	following	follow	VERB
ejpam-1857	222	6	curves	curve	NOUN
ejpam-1857	222	7	with	with	ADP
ejpam-1857	222	8	rank	rank	NOUN
ejpam-1857	222	9	exactly	exactly	ADV
ejpam-1857	222	10	6	6	NUM
ejpam-1857	222	11	.	.	PUNCT
ejpam-1857	222	12	y2	y2	X
ejpam-1857	223	1	=	=	NOUN
ejpam-1857	224	1	x3	x3	ADJ
ejpam-1857	224	2	−	−	PROPN
ejpam-1857	225	1	3546380914044004/81758650118401x2	3546380914044004/81758650118401x2	NUM
ejpam-1857	225	2	+	+	CCONJ
ejpam-1857	225	3	347413135010271276960000/739265720544437643649x	347413135010271276960000/739265720544437643649x	NUM
ejpam-1857	225	4	,	,	PUNCT
ejpam-1857	225	5	y2	y2	PROPN
ejpam-1857	226	1	=	=	NOUN
ejpam-1857	226	2	x3	x3	ADJ
ejpam-1857	226	3	−	−	PROPN
ejpam-1857	226	4	23507162841329764/648833431339681x2	23507162841329764/648833431339681x2	NUM
ejpam-1857	227	1	+	+	CCONJ
ejpam-1857	227	2	5359835105795753404473600/16527220769271504425329x	5359835105795753404473600/16527220769271504425329x	NUM
ejpam-1857	227	3	,	,	PUNCT
ejpam-1857	227	4	y2	y2	PROPN
ejpam-1857	228	1	=	=	NOUN
ejpam-1857	228	2	x3	x3	VERB
ejpam-1857	228	3	−	−	PROPN
ejpam-1857	228	4	3546380914044004/81758650118401x2	3546380914044004/81758650118401x2	NUM
ejpam-1857	229	1	+	+	CCONJ
ejpam-1857	229	2	347413135010271276960000/739265720544437643649x	347413135010271276960000/739265720544437643649x	NUM
ejpam-1857	229	3	,	,	PUNCT
ejpam-1857	229	4	y2	y2	PROPN
ejpam-1857	230	1	=	=	NOUN
ejpam-1857	230	2	x3	x3	ADJ
ejpam-1857	230	3	−	−	PROPN
ejpam-1857	230	4	23507162841329764/648833431339681x2	23507162841329764/648833431339681x2	NUM
ejpam-1857	231	1	+	+	CCONJ
ejpam-1857	231	2	5359835105795753404473600/16527220769271504425329x	5359835105795753404473600/16527220769271504425329x	NUM
ejpam-1857	231	3	.	.	PUNCT
ejpam-1857	232	1	references	reference	NOUN
ejpam-1857	232	2	139	139	NUM
ejpam-1857	232	3	references	reference	NOUN
ejpam-1857	232	4	[	[	X
ejpam-1857	232	5	1	1	NUM
ejpam-1857	232	6	]	]	PUNCT
ejpam-1857	232	7	j.	j.	PROPN
ejpam-1857	232	8	cremona	cremona	PROPN
ejpam-1857	232	9	.	.	PROPN
ejpam-1857	232	10	mwrank	mwrank	PROPN
ejpam-1857	232	11	program	program	PROPN
ejpam-1857	232	12	,	,	PUNCT
ejpam-1857	232	13	available	available	ADJ
ejpam-1857	232	14	from	from	ADP
ejpam-1857	232	15	http://www.maths.nottingham.ac	http://www.maths.nottingham.ac	PROPN
ejpam-1857	232	16	.	.	PUNCT
ejpam-1857	233	1	uk	uk	PROPN
ejpam-1857	233	2	/	/	SYM
ejpam-1857	233	3	personal	personal	ADJ
ejpam-1857	233	4	/	/	SYM
ejpam-1857	233	5	jec	jec	NOUN
ejpam-1857	233	6	/	/	SYM
ejpam-1857	233	7	ftp	ftp	PROPN
ejpam-1857	233	8	/	/	SYM
ejpam-1857	233	9	progs/	progs/	NUM
ejpam-1857	233	10	[	[	X
ejpam-1857	233	11	2	2	X
ejpam-1857	233	12	]	]	PUNCT
ejpam-1857	233	13	d.	d.	PROPN
ejpam-1857	233	14	husemoller	husemoller	NOUN
ejpam-1857	233	15	.	.	PUNCT
ejpam-1857	234	1	elliptic	elliptic	ADJ
ejpam-1857	234	2	curves	curve	NOUN
ejpam-1857	234	3	,	,	PUNCT
ejpam-1857	234	4	springer	springer	NOUN
ejpam-1857	234	5	-	-	PUNCT
ejpam-1857	234	6	verlag	verlag	PROPN
ejpam-1857	234	7	,	,	PUNCT
ejpam-1857	234	8	new	new	PROPN
ejpam-1857	234	9	york	york	PROPN
ejpam-1857	234	10	,	,	PUNCT
ejpam-1857	234	11	1987	1987	NUM
ejpam-1857	234	12	.	.	PUNCT
ejpam-1857	235	1	[	[	X
ejpam-1857	235	2	3	3	X
ejpam-1857	235	3	]	]	PUNCT
ejpam-1857	235	4	m.	m.	NOUN
ejpam-1857	235	5	kuwata	kuwata	PROPN
ejpam-1857	235	6	and	and	CCONJ
ejpam-1857	235	7	j.	j.	PROPN
ejpam-1857	235	8	top	top	PROPN
ejpam-1857	235	9	.	.	PUNCT
ejpam-1857	236	1	a	a	DET
ejpam-1857	236	2	singular	singular	NOUN
ejpam-1857	236	3	k3	k3	VERB
ejpam-1857	236	4	surface	surface	NOUN
ejpam-1857	236	5	related	relate	VERB
ejpam-1857	236	6	to	to	ADP
ejpam-1857	236	7	sums	sum	NOUN
ejpam-1857	236	8	of	of	ADP
ejpam-1857	236	9	consecutive	consecutive	ADJ
ejpam-1857	236	10	cubes	cube	NOUN
ejpam-1857	236	11	,	,	PUNCT
ejpam-1857	236	12	indagationes	indagatione	NOUN
ejpam-1857	236	13	mathematicae	mathematicae	PROPN
ejpam-1857	236	14	,	,	PUNCT
ejpam-1857	236	15	11(3	11(3	NUM
ejpam-1857	236	16	)	)	PUNCT
ejpam-1857	236	17	,	,	PUNCT
ejpam-1857	236	18	419	419	NUM
ejpam-1857	236	19	-	-	SYM
ejpam-1857	236	20	435	435	NUM
ejpam-1857	236	21	,	,	PUNCT
ejpam-1857	236	22	2000	2000	NUM
ejpam-1857	236	23	.	.	PUNCT
ejpam-1857	237	1	[	[	X
ejpam-1857	237	2	4	4	X
ejpam-1857	237	3	]	]	PUNCT
ejpam-1857	237	4	b.	b.	PROPN
ejpam-1857	237	5	mazur	mazur	PROPN
ejpam-1857	237	6	.	.	PUNCT
ejpam-1857	238	1	rational	rational	ADJ
ejpam-1857	238	2	isogenies	isogenie	NOUN
ejpam-1857	238	3	of	of	ADP
ejpam-1857	238	4	prime	prime	ADJ
ejpam-1857	238	5	degree	degree	NOUN
ejpam-1857	238	6	(	(	PUNCT
ejpam-1857	238	7	with	with	ADP
ejpam-1857	238	8	an	an	DET
ejpam-1857	238	9	appendix	appendix	NOUN
ejpam-1857	238	10	by	by	ADP
ejpam-1857	238	11	d.goldfeld	d.goldfeld	NOUN
ejpam-1857	238	12	)	)	PUNCT
ejpam-1857	238	13	,	,	PUNCT
ejpam-1857	238	14	inventions	invention	NOUN
ejpam-1857	238	15	mathematicae	mathematicae	VERB
ejpam-1857	238	16	,	,	PUNCT
ejpam-1857	238	17	44(2	44(2	NOUN
ejpam-1857	238	18	)	)	PUNCT
ejpam-1857	238	19	,	,	PUNCT
ejpam-1857	238	20	129	129	NUM
ejpam-1857	238	21	-	-	SYM
ejpam-1857	238	22	162	162	NUM
ejpam-1857	238	23	,	,	PUNCT
ejpam-1857	238	24	1978	1978	NUM
ejpam-1857	238	25	.	.	PUNCT
ejpam-1857	239	1	[	[	X
ejpam-1857	239	2	5	5	X
ejpam-1857	239	3	]	]	PUNCT
ejpam-1857	239	4	k.	k.	PROPN
ejpam-1857	239	5	ono	ono	PROPN
ejpam-1857	239	6	.	.	PROPN
ejpam-1857	239	7	euler	euler	PROPN
ejpam-1857	239	8	’s	’s	PART
ejpam-1857	239	9	concordant	concordant	ADJ
ejpam-1857	239	10	forms	form	NOUN
ejpam-1857	239	11	,	,	PUNCT
ejpam-1857	239	12	acta	acta	PROPN
ejpam-1857	239	13	arithmetica	arithmetica	PROPN
ejpam-1857	239	14	78	78	NUM
ejpam-1857	239	15	,	,	PUNCT
ejpam-1857	239	16	pp	pp	ADJ
ejpam-1857	239	17	.	.	PUNCT
ejpam-1857	240	1	101	101	NUM
ejpam-1857	240	2	-	-	SYM
ejpam-1857	240	3	123	123	NUM
ejpam-1857	240	4	.	.	PUNCT
ejpam-1857	241	1	1996	1996	NUM
ejpam-1857	241	2	.	.	PUNCT
ejpam-1857	242	1	[	[	X
ejpam-1857	242	2	6	6	NUM
ejpam-1857	242	3	]	]	PUNCT
ejpam-1857	242	4	sage	sage	NOUN
ejpam-1857	242	5	team	team	NOUN
ejpam-1857	242	6	.	.	PUNCT
ejpam-1857	243	1	sage	sage	NOUN
ejpam-1857	243	2	software	software	NOUN
ejpam-1857	243	3	,	,	PUNCT
ejpam-1857	243	4	available	available	ADJ
ejpam-1857	243	5	from	from	ADP
ejpam-1857	243	6	http://sagemath.org	http://sagemath.org	PROPN
ejpam-1857	243	7	.	.	PUNCT
ejpam-1857	244	1	[	[	X
ejpam-1857	244	2	7	7	X
ejpam-1857	244	3	]	]	PUNCT
ejpam-1857	244	4	m.	m.	NOUN
ejpam-1857	244	5	schütt	schütt	PROPN
ejpam-1857	244	6	.	.	PUNCT
ejpam-1857	245	1	k3	k3	PROPN
ejpam-1857	245	2	-	-	PUNCT
ejpam-1857	245	3	surfaces	surface	NOUN
ejpam-1857	245	4	of	of	ADP
ejpam-1857	245	5	picard	picard	PROPN
ejpam-1857	245	6	rank	rank	PROPN
ejpam-1857	245	7	20	20	NUM
ejpam-1857	245	8	over	over	ADP
ejpam-1857	245	9	q	q	NOUN
ejpam-1857	245	10	,	,	PUNCT
ejpam-1857	245	11	algebra	algebra	NOUN
ejpam-1857	245	12	&	&	CCONJ
ejpam-1857	245	13	number	number	NOUN
ejpam-1857	245	14	theory	theory	NOUN
ejpam-1857	245	15	,	,	PUNCT
ejpam-1857	245	16	4(3	4(3	NUM
ejpam-1857	245	17	)	)	PUNCT
ejpam-1857	245	18	,	,	PUNCT
ejpam-1857	245	19	335356	335356	NUM
ejpam-1857	245	20	,	,	PUNCT
ejpam-1857	245	21	2010	2010	NUM
ejpam-1857	245	22	.	.	PUNCT
ejpam-1857	246	1	[	[	X
ejpam-1857	246	2	8	8	NUM
ejpam-1857	246	3	]	]	X
ejpam-1857	246	4	j.h	j.h	PROPN
ejpam-1857	246	5	.	.	PROPN
ejpam-1857	246	6	silverman	silverman	PROPN
ejpam-1857	246	7	and	and	CCONJ
ejpam-1857	246	8	j.	j.	PROPN
ejpam-1857	246	9	tate	tate	PROPN
ejpam-1857	246	10	.	.	PUNCT
ejpam-1857	247	1	rational	rational	ADJ
ejpam-1857	247	2	points	point	NOUN
ejpam-1857	247	3	on	on	ADP
ejpam-1857	247	4	elliptic	elliptic	ADJ
ejpam-1857	247	5	curves	curve	NOUN
ejpam-1857	247	6	,	,	PUNCT
ejpam-1857	247	7	undergraduate	undergraduate	ADJ
ejpam-1857	247	8	texts	text	NOUN
ejpam-1857	247	9	in	in	ADP
ejpam-1857	247	10	mathemathics	mathemathic	NOUN
ejpam-1857	247	11	,	,	PUNCT
ejpam-1857	247	12	springer	springer	NOUN
ejpam-1857	247	13	-	-	PUNCT
ejpam-1857	247	14	verlag	verlag	PROPN
ejpam-1857	247	15	,	,	PUNCT
ejpam-1857	247	16	new	new	PROPN
ejpam-1857	247	17	york	york	PROPN
ejpam-1857	247	18	,	,	PUNCT
ejpam-1857	247	19	1992	1992	NUM
ejpam-1857	247	20	.	.	PUNCT
ejpam-1857	248	1	[	[	X
ejpam-1857	248	2	9	9	NUM
ejpam-1857	248	3	]	]	PUNCT
ejpam-1857	248	4	t.	t.	NOUN
ejpam-1857	248	5	shioda	shioda	NOUN
ejpam-1857	248	6	.	.	PUNCT
ejpam-1857	249	1	on	on	ADP
ejpam-1857	249	2	the	the	DET
ejpam-1857	249	3	mordell	mordell	PROPN
ejpam-1857	249	4	-	-	PUNCT
ejpam-1857	249	5	weil	weil	PROPN
ejpam-1857	249	6	lattices	lattice	NOUN
ejpam-1857	249	7	,	,	PUNCT
ejpam-1857	249	8	commentarii	commentarii	PROPN
ejpam-1857	249	9	mathematici	mathematici	PROPN
ejpam-1857	249	10	universitatis	universitatis	PROPN
ejpam-1857	249	11	sancti	sancti	PROPN
ejpam-1857	249	12	pauli	pauli	PROPN
ejpam-1857	249	13	,	,	PUNCT
ejpam-1857	249	14	39(2	39(2	NUM
ejpam-1857	249	15	)	)	PUNCT
ejpam-1857	249	16	,	,	PUNCT
ejpam-1857	249	17	211	211	NUM
ejpam-1857	249	18	-	-	SYM
ejpam-1857	249	19	240	240	NUM
ejpam-1857	249	20	,	,	PUNCT
ejpam-1857	249	21	1990	1990	NUM
ejpam-1857	249	22	[	[	X
ejpam-1857	249	23	10	10	NUM
ejpam-1857	249	24	]	]	X
ejpam-1857	249	25	l.c	l.c	PROPN
ejpam-1857	249	26	.	.	PROPN
ejpam-1857	249	27	washington	washington	PROPN
ejpam-1857	249	28	.	.	PUNCT
ejpam-1857	250	1	elliptic	elliptic	ADJ
ejpam-1857	250	2	curves	curve	NOUN
ejpam-1857	250	3	:	:	PUNCT
ejpam-1857	250	4	number	number	NOUN
ejpam-1857	250	5	theory	theory	NOUN
ejpam-1857	250	6	and	and	CCONJ
ejpam-1857	250	7	cryptography	cryptography	NOUN
ejpam-1857	250	8	,	,	PUNCT
ejpam-1857	250	9	chapman	chapman	NOUN
ejpam-1857	250	10	-	-	PUNCT
ejpam-1857	250	11	hall	hall	PROPN
ejpam-1857	250	12	,	,	PUNCT
ejpam-1857	250	13	2008	2008	NUM
ejpam-1857	250	14	.	.	PUNCT