curve #1164

y² = x³ + x² − 3621624323910547045280x + 247531180677433957517892273758100

a-invariants: [0, 1, 0, -3621624323910547045280, 247531180677433957517892273758100]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/4ℤ
conductor (N): 104034096695931795120
discriminant (Δ): -23429251998509983442800459090756176772012672758021427375209987840000
Faltings height: 11.6119
naive height: 162.7013
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 23, 29, 37, 41, 137, 191, 557
regulator: 11348194.14970519083630702006996408532292227619385686
submitted by: David Renshaw
submitted at: 2026-09-23 21:38:04 UTC
last updated: 2026-09-23 21:38:04 UTC

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

(15026989060, 14017928591500470)
(323732430301/4, 176099483358436365/8)
(33809593765/4, 117989459575942815/8)
(4313759983645/4, 8946428798888827155/8)
(429279161047885/13924, 21116633831031785228325/1643032)
(-2886663947900/49, 5492609911683509670/343)
(-81560718260, 599389355032650)

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 21:38:04 UTC · history