curve #1147

y² = x³ + x² − 30682329992528199195694848280x + 505633877706014012922832139156234849705600

a-invariants: [0, 1, 0, -30682329992528199195694848280, 505633877706014012922832139156234849705600]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 1128310354841987032068083640
discriminant (Δ): 1738161114326106750283883934756093697745475788190052358222531735838813272941107600000000
Faltings height: 15.4307
naive height: 208.3941
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 23, 31, 41, 53, 61, 67, 109, 257, 263, 6857
regulator: 233557909.634031665003976922249867242089139925297950721
submitted by: David Renshaw
submitted at: 2026-09-23 21:37:34 UTC
last updated: 2026-09-23 21:37:34 UTC

Witness: 8 independent points log in to add more points →

(-32182203519760, 1208192014418530050000)
(-9538996003285, 892997494010708936850)
(-2235318896624560/121, -1374315135119784722685600/1331)
(-83224318109345110040/1912689, 3507618465482873224340109725200/2645248887)
(-144952105945360, -1381121698192868948400)
(-1258574332119440/9, -38767370545383228557200/27)
(11084743961228840/10201, -708060992417925152410731600/1030301)
(522607806485744, -11278514444719522094640)

Commentary

Obtained by computing a rational isogeny from curve#1108 and mapping all 8 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 21:37:34 UTC · history