curve #5739

y² + y = x³ − 1492926000006873895

a-invariants: [0, 0, 1, 0, -1492926000006873895]
c-invariants: [0, 1289888064005939045064]
rank (lower bound): ≥ 12
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 1320786987553495959933110666920909083
discriminant (Δ): -962853713926498554791237676185342721507
Faltings height: 6.1591
naive height: 97.2177
primes of bad reduction: 3, 223, 859109, 1154466611
regulator: 924327029380162260.55881088812214971930403183766122571763733999
submitted by: jesper-petersen
submitted at: 2026-10-24 18:09:43 UTC
last updated: 2026-10-24 18:09:43 UTC

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

(177355998761649985/124290092304, 52079620519490562076276673/43818223461590592)
(14155051822318057465/8525567556, 53255834638549710190405896601/787199754715704)
(221174797323339073/69155856, -104016792348282838959272897/575100098496)
(221495753975393089/4693346064, -104242549305831656705081281/321531752152512)
(14180633263126321345/346386394116, -53399687239581501361089227995/203864326711395336)
(14203471440287634265/530662713156, -53527239420229211959357145695/386569744001898696)
(198896537859350205313/34048345329216, 2794527715057375629149166647231/198675363637126964736)
(568852073124894442424084978661769/281256416602691614350418944, 12282528238941056503828530805291143701377797286619/4716867305507051365425820071149641039872)
(121417080612282229240377721/104946722698799592900, 253622967066460803607895594470267660519/1075111039488424087600801533000)
(163842838676984456281/17047121561856, -2095442485116996561216574461533/70384428258536042496)
(2519920934423111762447329/16153459305987600, -4000186928192274918434102858544194767/2053044263853219566376000)
(12751674321823736955294385/4393884682978802064, 44123062411053127901003136864424912585/9210283155281581708903624512)

Commentary

The rationally 3-isogenous partner of the Burkhardt-quartic Mordell specialization obtained from parameters (u,v,w)=(7/6,-5/6,3/6) at t=-11/336. PARI/GP 2.17.4 returned matching computational rank bounds [12,12] for this curve and the original model.

last edited by jesper-petersen at 2026-10-24 18:09:43 UTC · history