curve #5707

y² + y = x³ − 103109368800425885

a-invariants: [0, 0, 1, 0, -103109368800425885]
c-invariants: [0, 89086494643567964424]
rank (lower bound): ≥ 12
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 31243715072587803983534516506247361 ☆ record for Mordell curves, rank ≥ 12
discriminant (Δ): -4592826115670407185579573926418362067
Faltings height: 5.7137
naive height: 91.8723
primes of bad reduction: 3, 7, 8417091330647011
regulator: 295184372145515560565.75310400626972332757104263230351092881530
submitted by: jesper-petersen
submitted at: 2026-10-24 14:08:14 UTC
last updated: 2026-10-24 14:08:14 UTC

Witness: 12 independent points log in to add more points →

(229279798070360305/39280068864, -109757971990604842513967305/7784995408293888)
(236694501208362546721/333489540796416, 3071873008454642056758643418447/6090084613203746881536)
(1888367068417562794937145963418280878913193130751905/106906966706109971784134816450302684631387364, 82058876091975922546846676944387113991753721772969428447588016527229125530195/1105373404314912416694191354192227522477764321695602341575138646888)
(40637628440769883150479012246029905/3644888749440703432068608064, 8191744092674066548770476517022115958977276795401433/220052555148086067992525172793642151296512)
(15387393650069447958476916313057/29464101234247945215354244, 31715796080147969722128425353060700431965913507/159933630931369994455009887194865294472)
(38533404412998754595132744577297985/680870770238514631630561344, 7564077104728562757714159729150421161360796620008545/17766301808377846068610311548713615523328)
(326412495344025919495824145/240891101082838277604, 5773762860689161037303562372118947770995/3738790539562744415427090662808)
(3320614516695425870714929/101742672361297984, -6050998555087573731811616353516908119/32452989619762461527515648)
(1071283909978303107807739067809/343636714821305688776464, 1106920995164798940011282786024054757621890351/201441681403561696152369651802762688)
(5876468182701682235591746912979700220997545/11269200209313895503648987862427040196, 7440993345897938449538896745925633718424105921565810684262010471/37830287514414445343812779530653972733056910981367842744)
(166395756527608029680418290673641207809193617/86830993796597430962724473576104773904, 2130630788608058794686335017696825320768281093046098193527068173913/809118548965637748533924477340612259436780748960173814208)
(115816617684022626650831907253539238559796332632404265/22308092747298957977475762811496432717664741924, -39400002994568814862505326511410417160225729689646356358137749023390950948898863/3331913070428923702821356142251789672008994536445704432858352202250968)

Commentary

The rationally 3-isogenous partner of the Mordell curve obtained from the Burkhardt quartic with parameters (u,v,w)=(2/3,-4/3,2/3), specialized at t=419/164.

last edited by jesper-petersen at 2026-10-24 14:08:14 UTC · history