curve #1163

y² = x³ + x² + 393463057617825505x − 72394612235805314278716750

a-invariants: [0, 1, 0, 393463057617825505, -72394612235805314278716750]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/4ℤ
conductor (N): 892331075985258840
discriminant (Δ): -6162552357420272961294560401623129531548558729875575600
Faltings height: 9.2005
naive height: 133.1549
primes of bad reduction: 2, 3, 5, 7, 17, 19, 29, 53, 71, 73, 313, 1319
regulator: 573339.52012222038967873733037658666980912652469748
submitted by: David Renshaw
submitted at: 2026-09-23 21:38:01 UTC
last updated: 2026-09-23 21:38:01 UTC

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

(11377784475, 1215443878690125)
(20517117475, 2940191186843925)
(56107034034850/841, 420286179870146333100/24389)
(49522392313/16, 11231373061214229/64)
(400907179, 12238626827805)
(1090958583695602/196249, 36255117897811644511992/86938307)

Commentary

Obtained by computing a rational isogeny from curve#1111 and mapping all 6 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 21:38:02 UTC · history