curve #312

y² + xy + y = x³ − x² − 6568019580778599489585141337989692763602574718655x + 6357048702469147640200079650140599226292653501480652916481851397620447847

a-invariants

[1, -1, 1, -6568019580778599489585141337989692763602574718655, 6357048702469147640200079650140599226292653501480652916481851397620447847]

c-invariants

[315264939877372775500086784223505252652923586495425, -5492490078933343561132867399029248283339362874888755114066682669387927710625]

rank (lower bound)

≥ 19

torsion subgroup

ℤ/2ℤ

conductor (N)

565363502863493379565569563649307108299188776386222378070363275106421667133549554450

discriminant (Δ)

675555591393920126210578335859174012733377876129576927329773074052725622143925022750267503150613919256894560960666950209537112482403220443264000000

Faltings height

26.9627

naive height

348.8325

primes of bad reduction

2, 3, 5, 7, 13, 17, 19, 47, 61, 71, 79, 83, 101, 127, 137, 229, 241, 293, 311, 523, 541, 827, 1009, 1753, 3331, 4091, 6637, 9007, 16069, 44263, 52609, 67157

regulator

9932244623647556444960930996.536406068648448247486208204263816285117397787804

submitted by

7fff-zip

submitted at

2026-08-24 00:23:57 UTC

later contributions

rank ≥ 15 → ≥ 19 · David Renshaw · 2026-09-02 19:09:05 UTC

Witness: 19 independent points log in to add more points →

(1899281807852383583281523, 856594138321723281977125415211154230)
(1986644172969550454297619, 1072160825764798064703099415629555990)
(272033437632697930156443795, 4486567741374453996533205302226083011478)
(-73812922834259315731199078177667772822524301/35487061367080894081, 701760471319037145575304587030602480113522596985733764773681362890/211400035691578538825217788479)
(-3927187767058422894394186604590595226892741/1539769725771925561, 4877843076530640854613063041455010708637094178667651886192296990/1910661113303723032069064941)
(-1379469481343674515359008391245128809/476344291329, 343353156490727528348171283886941951282816243230945030/328761873956575233)
(134729047327530661756060101/25, 1411330887439593853235479471234166597434/125)
(73956250939559691477602244623/37636, 7440245430807044698527020029945896376490415/7301384)
(830793851953810726479392048429/143641, 692591951097164080309050564516308392473126360/54439939)
(180854862119352253161517984249697/108014449, 260511941958570959871331446642478485548227326972/1122594168457)
(573069003295601293644444034798199/1907161, 13718121089796567437611628542807734825088078010370/2633789341)
(267893840271520846495661485953911955/207608720881, 16492262626919596964764921351588020331572391080530198/94595045190939721)
(3821986387736518845234207009880766699/1758962935081, 3571956568922634123068914143789971753810589794102731890/2332840423317591979)
(61088598901691622750773457689296849259/32549888973001, 148664262708786375589192736275700644681489711698052561170/185705286613102928251)
(62897219236830187109680220822998179931/210557018999889, 6424658587759947038223786820844122894975074773416404566170/3055305100430466325287)
(10510452475024226027374308869767757423001859/255290295969285201, 34010179064696150792968074723430580552211722473010843913014874470/128988638951500856768067801)
(4155100990995848311306968282657694405233292611/3672460189087728729769, 136140361456226024969533946302837075564652577856632001911107316164470/222554123250189687936815390056853)
(4808487952566005874507210289295926305526194112331/4745410311994898429046409, 8905367989354703625912085371452793646493479229959140225648741207586516590/10337384126367975930596225696661735077)
(10778558758427320881675942446210189105048914179369/6554505074799611532809761, 951815842838653131467287993135740878144334987059084577329062215300009390/16780691607221944670986272889179239759)

Commentary

2-isogeneous to the curve [1, -1, 1, 31368015812338065133318565292206590792820353345, 302038802698566087335643188429543498624522041683874493555186062568159847] found by Elkies (2009)

last edited by dujella at 2026-09-23 19:14:50 UTC · history