curve #5696

y² = x³ − 73779374709082522x

a-invariants: [0, 0, 0, -73779374709082522, 0]
c-invariants: [3541409986035961056, 0]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 348377352477645163379230656376350976
discriminant (Δ): 25703063228606300850778943499103199656627355219241472 ☆ record for quartic twists, rank ≥ 11
Faltings height: 8.7460 ☆ record for quartic twists, rank ≥ 11
naive height: 128.1332 ☆ record for quartic twists, rank ≥ 11
primes of bad reduction: 2, 53, 89, 509, 821, 1669, 11213
regulator: 36752222060250.1833128625351461424073942764072683229531255263
submitted by: Michael Rubinstein
submitted at: 2026-10-24 13:20:50 UTC
last updated: 2026-10-24 13:20:50 UTC

(-716439145019689310553/18553858441561, 133523587872660528307180413816303/79919242374490241059)
(-540307285610747561/3434546025, 559018352356363977060987863/201281569795125)
(-348728370500000/1830609, 6619040999088603841000/2476813977)
(-1594070093713/6889, 1237304431990455603/571787)
(-53129485625/6241, 390549487961444175/493039)
(2329168721857/4624, 2993581265230828383/314432)
(43701244477578/1681, 288879684195521441526/68921)
(715161186531226/1034289, 17587737122838319160542/1051871913)
(1256260476802682/2968729, 34142977071121437479298/5115120067)
(1962974012227519833/364886404, 2746739951475831013959625011/6970060089208)
(208793041813916945537/456770519104, 2426566744892931756601863087423/308707441795400192)

Rank 11, proved: 11 independent points listed, matching the 2-descent upper bound. Torsion Z/2. Member of the family y^2 = x^3 + a x.

last edited by Michael Rubinstein at 2026-10-24 13:20:50 UTC · history