curve #1135

y² = x³ + x² − 273244185676276338095x − 1728848290289983362878722359600

a-invariants: [0, 1, 0, -273244185676276338095, -1728848290289983362878722359600]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
conductor (N): 104034096695931795120
discriminant (Δ): 14456116515114485104741059623665082731055450521922541111886800
Faltings height: 10.5722
naive height: 152.7843
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 23, 29, 37, 41, 137, 191, 557
regulator: 1452568851.162264427047298568955402921334051352813678
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:08 UTC
last updated: 2026-09-23 21:37:08 UTC

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

(-73641212573375/7744, -66831197917789048065/681472)
(-231170378234384/24025, 362469290922923614536/3723875)
(1158549043644400/34969, 33075834757354026203160/6539203)
(-23299990715956880/2405601, -355340248761564245727080/3731087151)
(-248406842384/25, -8993959156789464/125)
(2652113546905/64, -3902910872463518685/512)
(2010014284101169681/43264, -2849697019098265295032786695/8998912)

Commentary

Obtained by computing a rational isogeny from curve#1089 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 21:37:09 UTC · history