curve #2865

y² + xy + y = x³ + x² − 539991948063654653632810479817028764587900x − 117549437565021027150808486136019745497474050697277533028184115

a-invariants: [1, 1, 1, -539991948063654653632810479817028764587900, -117549437565021027150808486136019745497474050697277533028184115]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/4ℤ
conductor (N): 4738696264040550563800319661390
discriminant (Δ): 4107925243910968756555577971267932138331930160031355212466993630352132020561196400611012077776951360146443969310433542006250000
Faltings height: 23.0065
naive height: 299.8907
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 29, 41, 71, 113, 193, 1889, 171679, 818231
regulator: 58756461.329013825061675423983868403898822493485117
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:25:56 UTC
last updated: 2026-09-26 20:25:56 UTC

Witness: 6 independent points log in to add more points →

(-536374889645973177161923/1369, -294813127701864286112391941117879751/50653)
(710424392482594315958683/676, -383319196366071892733652175741354367/17576)
(-47478121212002788565577233/85849, -86909036655977036213915599321157458769/25153757)
(1906115601627742684914024114439543/1473053125636, -65698404381948908979385410503893129616495628350723/1787835740265659384)
(5608265243269457053003, -416231636017207173248588032741417)
(7551440082504255508877027190787/6171359364, 15737970533326346107424127492587254684910981371/484809648917112)

Commentary

2-isogenous to curve#1897 via the rational 2-torsion point x=290566547041438749261. Independent witness points are the images of that curve's witnesses under the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:25:56 UTC · history