curve #4771

y² = x³ − 75678650779410795595704225x

a-invariants: [0, 0, 0, -75678650779410795595704225, 0]
c-invariants: [3632575237411718188593802800, 0]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 7330890475253778201214555373723049650024030314048800
discriminant (Δ): 27739595008941975970493171735842303123910304344963255535048054320147350809000000
Faltings height: 13.9332
naive height: 190.3792
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 109, 173, 353, 1009, 1181, 1609, 4157
regulator: 7499001827716.4815733960709939379629143001266898336774597872154
submitted by: David Renshaw
submitted at: 2026-10-20 21:29:34 UTC
last updated: 2026-10-20 21:29:34 UTC

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

(-17513163291615/4, 125835143053829807625/8)
(-8673709284240, -1965560103038231700)
(-8452322325135, 5984281772072580600)
(438368987131735, 9176439379286807197900)
(-1807788823809540/361, -109184767171932965949150/6859)
(-167862964815/4, -14256702039745230975/8)
(-18925708394113521/2209, 458447881421645369846592/103823)
(-8601183539025, -3821953107932836800)
(8700196120924, 358371642744720682)
(103108802726479/9, -681277904115856625908/27)
(67438142542225/4, 474388125118648406575/8)
(60954743269204015/169, -15044747346590088784772500/2197)

Commentary

Quartic twist y^2 = x^3 + a x with a = -75678650779410795595704225. 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 21:29:34 UTC · history