curve #307

y² + xy = x³ − 1474705521041100338570986911092165596800135x + 689016723668927095487122547447296652854367658972300541602585225

a-invariants

[1, 0, 0, -1474705521041100338570986911092165596800135, 689016723668927095487122547447296652854367658972300541602585225]

c-invariants

[70785865009972816251407371732423948646406481, -595310449249953010500980059791979267290550768409666303867603244121]

rank (lower bound)

≥ 18

torsion subgroup

ℤ/2ℤ

conductor (N)

113867782999071766513257245176182365522858404266111886760688305190 ☆ record for ℤ/2ℤ torsion, rank ≥ 18

discriminant (Δ)

166587152111400413731938961061747813215845690391267294429737525157756851883138882108010598181179320356424466059369807216640000

Faltings height

22.9836

naive height

302.9047

primes of bad reduction

2, 3, 5, 7, 11, 17, 19, 23, 71, 127, 151, 179, 241, 281, 449, 503, 1223, 1361, 1399, 2017, 2083, 2293, 2843, 5651, 74923, 436549

regulator

170881194038617975643152.69941779064453316329287758069709759128083041511231

submitted by

7fff-zip

submitted at

2026-08-24 00:23:17 UTC

later contributions

rank ≥ 15 → ≥ 18 · David Renshaw · 2026-09-02 19:07:06 UTC

Witness: 18 independent points log in to add more points →

(688026471287925627882, 280270155855420880369948469961)
(1920603132489760518390, 70294046253440894558287052273445)
(6224496356871005112630, 482362740560791652609244112713765)
(-6568200808203233585856636180258330/5791208559049, 418712317394027810613867130100862094323187144278585/13936502858891505157)
(-85865982697616384965827153210/61889689, 3908168356857862855582783172982510263148975/486886183363)
(205957641842136395549190/289, 203099586524940527641301708521845/4913)
(4730081366485313902607670/6889, 228542270776522233480187196075882055/571787)
(11035607107999604409970330/23409, 35559346195761543789312341621450685205/3581577)
(149814166625115251622794170/221841, 109727998398671263942933912301974284815/104487111)
(379658655163433571885453150/677329, 3459676170376871902239639410956478185755/557441767)
(3382696334484158237159967062/5013121, 12101727263164295476473400572359379464699/11224377919)
(49108627265451539372015730696/71250481, 74671922092507506108917362740098695308597/601425310121)
(121569508797092261098235238070/892276641, 590368288997057839132580950896485943481257515/26653195543311)
(772038729206856510119284176165/1156272016, 56054439894098773613457710060770913764070685/39317873632064)
(3623082257776861607646263836410/4541007769, 1379052513524016167502360628685726608595555835/306004890529603)
(29706007003945342560456287904600/20943588961, 115447680553194400185523816746990916290859319715/3030935250846959)
(12970328663837241363442993836437430/18850905948169, 22766786707038492160435625994181716292943086744455/81846165962240081947)
(15624902786915746905214005579828438/52864198183729, 6419719425295421244163122540351936855565112920374381/384363796277698587433)

Commentary

2-isogeneous to the curve [1, 0, 0, -1718612993735110076283239582307203184558215, 445677626128337788660554947611110167094039534522619288819463817] found by Elkies (2009)

last edited by dujella at 2026-09-23 19:17:32 UTC · history