curve #4764

y² = x³ − 10356583068229284172x

a-invariants: [0, 0, 0, -10356583068229284172, 0]
c-invariants: [497115987275005640256, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 156696585608668361973767378558845472
discriminant (Δ): 71093427524589409801900282724050219961671343907834538012672
Faltings height: 9.9821
naive height: 142.9661
primes of bad reduction: 2, 11, 13, 23, 29, 31, 37, 43, 53, 59, 67, 71
regulator: 938940960.60455470219321491488405399593028412983239966495
submitted by: David Renshaw
submitted at: 2026-10-20 21:29:06 UTC
last updated: 2026-10-20 21:29:06 UTC

Witness: 9 independent points log in to add more points →

(-324610088356879/1196836, 69147424517775678411753/1309338584)
(-6369046695882/2209, 7969476318404041116/103823)
(-707671855098/289, 507670107191439972/4913)
(-535100097674/1089, 2533560906661190108/35937)
(152382607894/9, 58400024944943924/27)
(215696318436/49, 68349921284772492/343)
(709623495097/16, 596205081499141587/64)
(958432317337/144, 821329086991786307/1728)
(1467313461238/441, 451372032319111084/9261)

Commentary

Quartic twist y^2 = x^3 + a x with a = -10356583068229284172. 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 21:29:06 UTC · history