curve #4767

y² = x³ + 303523956992981037543615x

a-invariants: [0, 0, 0, 303523956992981037543615, 0]
c-invariants: [-14569149935663089802093520, 0]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 327561928777518231048733993255374727655208064800
discriminant (Δ): -1789612070090496502451787786422100724771713992376875352182756335432536000
Faltings height: 12.5535
naive height: 173.8228
primes of bad reduction: 2, 3, 5, 7, 11, 19, 103, 151, 181, 631, 661, 977, 4019
regulator: 3196751624401.5216737973163515733444237502143462307635738118702
submitted by: David Renshaw
submitted at: 2026-10-20 21:29:18 UTC
last updated: 2026-10-20 21:29:18 UTC

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

(423155124540, 451893775348988190)
(165565843839, 234076267550441448)
(40659591856240, 259289175252550021460)
(1058045903140615/52441, -940395921828290231705360/12008989)
(665743024180521/256, -17558737699302388438101/4096)
(1736744050881, 2401176710640168216)
(2409106140, -27041364802479510)
(490917930457735/729, 14053432975688150269280/19683)
(31377338839/9, -878326015641980752/27)
(61671958230289/81, 597812525663662594648/729)
(3123847890625, -5606424888990695000)
(2102828092820665/1296, 101835630353252720291885/46656)

Commentary

Quartic twist y^2 = x^3 + a x with a = 303523956992981037543615 (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 21:29:18 UTC · history