curve #4765

y² = x³ + 113679909718764574217823x

a-invariants: [0, 0, 0, 113679909718764574217823, 0]
c-invariants: [-5456635666500699562455504, 0]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 45948877773036317495588418797306267208003055392 ☆ record for quartic twists, rank ≥ 12
discriminant (Δ): -94022356984511702308503105325196023260416262965891661652406385767729088 ☆ record for quartic twists, rank ≥ 12
Faltings height: 12.3080 ☆ record for quartic twists, rank ≥ 12
naive height: 170.8766 ☆ record for quartic twists, rank ≥ 12
primes of bad reduction: 2, 3, 7, 43, 47, 71, 73, 197, 199, 577, 2141, 3557
regulator: 1664736199910.2498982879759155485437612546515592832197605539342
submitted by: David Renshaw
submitted at: 2026-10-20 21:29:10 UTC
last updated: 2026-10-20 21:29:10 UTC

Witness: 12 independent points log in to add more points →

(104762203270617/4, -1072365712734453758613/8)
(10655517447792, 34799990290591352148)
(24989725443036/169, 311034855129139282842/2197)
(757689992791239/3721, 40343337019608251813064/226981)
(7445251230832, -20335923846951178348)
(733498061692, -691390038441984398)
(675107014724289/7921, 71574546853697268494736/704969)
(652378328741199/1225, 19721571715238548411332/42875)
(127167789084, 128502666601497594)
(289395375129/4, -742032218204891019/8)
(1151258506176, -1287147447859797432)
(264081393534006951/570025, -167843273951936378691040824/430368875)

Commentary

Quartic twist y^2 = x^3 + a x with a = 113679909718764574217823 (2-isogenous model of the curve listed there). Source: Aguirre, Castaneda, Peral, 'High rank elliptic curves with torsion group Z/(2Z)', Math. Comp. 73 (2004) 323-331, doi:10.1090/S0025-5718-03-01547-3, https://www.ams.org/mcom/2004-73-245/S0025-5718-03-01547-3/.

last edited by David Renshaw at 2026-10-20 21:29:10 UTC · history