curve #3985

y² + xy = x³ + x² − 613309634524992782x + 184868945051812476207225060

a-invariants: [1, 1, 0, -613309634524992782, 184868945051812476207225060]
c-invariants: [29438862457199653561, -159726768745557447872039853485]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ
conductor (N): 686044836958354728450143389446
discriminant (Δ): 260458878016993866992898967860704696649784248398352
Faltings height: 8.8321
naive height: 134.4865
primes of bad reduction: 2, 3, 11, 17, 31, 101, 157, 199, 347, 487, 593, 2351
regulator: 288944231484399.614624938973452184543366475603273522971798639
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:39:43 UTC
last updated: 2026-10-05 04:39:43 UTC

Witness: 11 independent points log in to add more points →

(195646059049/256, -52218751204640083/4096)
(558815915714749/1229881, -97339639877793743717/1363938029)
(31354794783096/69169, -259805569175373582/18191447)
(4651068288696/10201, 137870294263773990/1030301)
(3299431628056/7225, 99265919790693666/614125)
(7434901164178215651648061/13642478973846529, -5631610482506117227588055419204110876/1593455500401264832199167)
(144678832271416/231361, -753819451289539471266/111284641)
(69764959984861/145161, 58514659398909635555/55306341)
(22944435061/49, -204291996614644/343)
(776743839583648216/1711725129, -3171676775558318601968366/70819203762117)
(86614103340799952929/190969000000, -2717904067156517833138182717/83453453000000000)

Commentary

Unlisted rational-isogeny neighbor of curve#3625 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 11: 11 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 04:39:43 UTC · history