curve #4001

y² + xy + y = x³ + x² − 1385318294350713638263921710913x + 627584819845954744074343175234538294770980031

a-invariants: [1, 1, 1, -1385318294350713638263921710913, 627584819845954744074343175234538294770980031]
c-invariants: [66495278128834254636668242123825, -542233284346905397594818469659550861693942675625]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 84518160226589403767572961475020918466832050
discriminant (Δ): 29461828723379382017989879893344413944855161380284598234594602571290086867047056000000
Faltings height: 15.8315
naive height: 219.8241
primes of bad reduction: 2, 3, 5, 7, 11, 19, 23, 47, 67, 71, 79, 97, 167, 173, 211, 313, 367, 431, 743, 1621, 4643, 5791
regulator: 116763676981644047990.708409496982368846649434500638169333952583452
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:43:20 UTC
last updated: 2026-10-05 04:43:20 UTC

Witness: 14 independent points log in to add more points →

(41753623735616345446685/17901361, -7606316577844376990633683858208538/75740658391)
(39765160348884261709765/25979409, -6058341005182479430596346720972264/132417047673)
(70039880621885683135/103041, 144084892124013760912374752/33076161)
(36403392636091415/49, 996815228626896036198576/343)
(4356706679230119379/6084, 790204191377589435759333631/474552)
(51956477363475566567081/76440049, -350729293690569369742167568134/668315348407)
(330672885038541584671/18225, 6000743697815370752938964568176/2460375)
(33772672603345395432098331/49686639025, -31146712807186867938424710057893944/11075400271867625)
(8122055524038660165891435/10615386961, 4314045705437292871985784165124643012/1093713933978791)
(1701044880520061589636321/2497500625, -8747762039676417999709719691833056/124812593734375)
(33175764714928730285044328090985/48698193123642241, 828150307259841052371389182726982209185771952/10746555142240461138859711)
(8500152780311546645791417394915/12504210451239529, -11501427945562236006167743482563375370586044/1398248659534279043211083)
(55154896167104140664326325335535/79549866579047881, -14024946665813246912815836365017268168376930672/22436710452622950998774779)
(52445513093152773139980391310661935/35358173781415671481, 9009083800088645827529894724702470426623089262716072/210249387357781437416487176621)

Commentary

Unlisted rational-isogeny neighbor of curve#3643 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 14: 14 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:43:20 UTC · history