curve #3961

y² + xy + y = x³ − x² − 2175082126494811020121091x − 330978588667181391045789339126477334

a-invariants: [1, -1, 1, -2175082126494811020121091, -330978588667181391045789339126477334]
c-invariants: [104403942071750928965812353, 285965500608914539602884868185622571935]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 2401768163536149420691858179
discriminant (Δ): 611253367769755643367231849217796867438962402300183543227254655822772970409
Faltings height: 13.0413
naive height: 179.7309
primes of bad reduction: 3, 7, 11, 13, 23, 43, 47, 149, 991, 1153, 27031, 1246261
regulator: 455450380448.21172913142946134293863822829417355937898019
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:34:56 UTC
last updated: 2026-10-05 04:34:56 UTC

Witness: 9 independent points log in to add more points →

(-10642720428605721/14884, 1682683133044167333383075/1815848)
(-139024865197974891/100489, -5534663801616367781518675/31855013)
(21573413522060004589/1575025, 99613397327204672852585788827/1976656375)
(-4925766613921277968731/6318978064, 474106984472455408901908403924239/502308204263488)
(-1680276992110141151/1721344, 2096901619638400224799050175/2258403328)
(20438264785045981/289, -2921267976654095336891635/4913)
(4232946352706235159909/1579188121, -227114608877838799030210913625365/62755356740419)
(-4993903476966806499/7491169, -18597726488538387968184815435/20503329553)
(-31695810175470706331/23464336, -42882719888848147218063540535/113661243584)

Commentary

Unlisted rational-isogeny neighbor of curve#3675 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 9: 9 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 04:34:56 UTC · history