curve #3616

y² + xy + y = x³ − 45141198526458352945133705095533926x + 2062218282541741516059786146320833327182658297331748

a-invariants: [1, 0, 1, -45141198526458352945133705095533926, 2062218282541741516059786146320833327182658297331748]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 1276790687761151076804421275353458888576233831160016970956831646271429350
discriminant (Δ): 4049884909363351079009747519467402173327926713990480440831144208879352817953741157862530932627482073062500
Faltings height: 18.9624
naive height: 250.9989
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 23, 31, 37, 41, 43, 53, 107, 109, 2777, 371682981688163, 1160163369385000545668134854830921
regulator: 1302173947120592844163397163362.3637063438983675814410249910663982650829258653975
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:20:11 UTC
last updated: 2026-09-29 19:20:11 UTC

Witness: 21 independent points log in to add more points →

(-10616214366335638465753004756106957/58056219649707889, 906805792640893168929398558413357022370332620370497/13988563858933872218471287)
(-47401850193040061684735668461391/911586271245169, 1798274627860929242341698392094860037131546148513/27523056643948238845303)
(-6860629016046081788435826033/96596018401, 2103690396379225937151053525694944721902909/30021945923012399)
(-17789168336836090738908727947/256955720464, 9076257229721264409604324913622836839562707/130252910348965312)
(-41061721384451549602756974/335073025, 465248365516035982901596766466019078061/6133511722625)
(-33146368578388447326768/143641, 752714905442465815286748543242929/54439939)
(140238058970121037429897194/5686215649, 13312309115450245093388225413264537105931/428780463444143)
(363166183074488964798358067/166384201, 6888652215342413815834703370178631465109/2146189808699)
(742389017711185723529771052/1621431289, 18161689432312489049308987176368122266421/65290173714163)
(42421222474004843018893457417/179321759296, 5162695932982202376358863527068439077086009/75936309478521344)
(12137434957966143654748586706/51388809481, 788047466012043995332170679363725955018976/11649380610057371)
(17378357962853082980757131254/3200956929, 2289196619657494124519483718976122900052936/181100540172033)
(17727580210753660101691959168/422893991809, 4292060718369669519353091636478471998829397/275009231555368127)
(195237226174777415604941608653/5233607893264, 248323356586200900581205256111606838100859807/11972966646283198912)
(6589282642946879596055592917208/14905182082729, 15061346667831792412872597021720754111703921537/57544779285979753067)
(373652459774951885300610168509243/1058961132472489, 5975121453110258851495017627063288131378027253447/34460377482240743258987)
(469992367080322064122327860161913/1978612115817649, 6060802988505069201556481201754593560117606444067/88011818840669525034857)
(1624001918606182552653283429609818/2955639883445209, 60794270709564229248687887453746005256674420357553/160685717908485893116573)
(3213133030828216408780295039245092/16220278456346449, 61747610322577858415261076708552232257542808235372/2065796124422008140648793)
(45573406533310852044341923169065218/3262266128295001, 9727856815112443848442936942876037178461907793956967/186328411012495479161251)
(166260132949648858647422792394664167/723585178872886201, 38047069563414181352062784243645578749739028090672441/615509122468177238407744301)

Commentary

Specialization at T = -293/64 of the rank-17 fibration of X1008, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #751 (same T), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:20:11 UTC · history