curve #1134

y² = x³ − x² + 72827732429x + 42994416559630900

a-invariants: [0, -1, 0, 72827732429, 42994416559630900]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
conductor (N): 1321186210228868621060280
discriminant (Δ): -823282724219542148027966839348392240
Faltings height: 5.5719
naive height: 90.1228
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 23, 29, 31, 37, 8420798017
regulator: 131996664.7939473882806577724748567563409267747789741
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:06 UTC
last updated: 2026-09-23 21:37:06 UTC

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

(-2939134000/11881, 128456738510970/1295029)
(7058563149/16129, 815971226925059/2048383)
(30186301629/19321, 5351344815100447/2685619)
(49520329349/39601, 11391342109618563/7880599)
(419139209904/218089, 274829393063178526/101847563)
(47584515021/93025, 13128003940452269/28372625)
(4033592343869/1974025, 8191545229924947753/2773505125)

Commentary

Obtained by computing a rational isogeny from curve#26 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 21:37:06 UTC · history