curve #1178

y² + xy + y = x³ − 466361339658083416048x − 1466755953607436274826469298322

a-invariants: [1, 0, 1, -466361339658083416048, -1466755953607436274826469298322]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/6ℤ
conductor (N): 783917988850972810230
discriminant (Δ): 5562144759605538220635705466429765263015593914775496606636858000
Faltings height: 10.9246
naive height: 154.3881
primes of bad reduction: 2, 3, 5, 11, 19, 31, 37, 53, 103, 383, 487, 107053
regulator: 40576862.67827447377479229407212767722663765023147677
submitted by: David Renshaw
submitted at: 2026-09-23 21:38:30 UTC
last updated: 2026-09-23 21:38:30 UTC

Witness: 7 independent points log in to add more points →

(-8185497083, -1342454766599125)
(-3545927963, 377285039973035)
(3071224813806433/16129, -169086943278163373458735/2048383)
(1864353018322, -2545441953737743513)
(-26728336708574/1849, -119367940705874468155/79507)
(-2229098332709726/156025, -93057369581579247073358/61629875)
(-1043832400164926/201601, -81422561709074061018665/90518849)

Commentary

Obtained by computing a rational isogeny from curve#1121 and mapping all 7 independent witness points to the global minimal model. Isogenies preserve the Mordell-Weil rank; the image points are independently certified by the site verifier. Torsion computed exactly with PARI/GP elltors. Conductor is unchanged by the isogeny.

last edited by David Renshaw at 2026-09-23 21:38:30 UTC · history