curve #4769

y² = x³ − 1645077324548360946504525x

a-invariants: [0, 0, 0, -1645077324548360946504525, 0]
c-invariants: [78963711578321325432217200, 0]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 3464037636791287415327029772074929316884282208800
discriminant (Δ): 284930488383371900247961754132105193825609265627970786467479000309741000000
Faltings height: 12.9760
naive height: 178.8931
primes of bad reduction: 2, 3, 5, 11, 19, 31, 43, 47, 61, 67, 73, 197, 251, 271, 419
regulator: 453739398352.91129366342155065430200215001782034416117053915068
submitted by: David Renshaw
submitted at: 2026-10-20 21:29:26 UTC
last updated: 2026-10-20 21:29:26 UTC

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

(6452826676905, 16064671526784139350)
(-152160578405, 496782615891022450)
(-18280131891975/16, 39874092186704525325/64)
(-1258605889845, 277056914531579850)
(1315287167920, -334173366644474500)
(1465715472780, -858844155457885350)
(58768107612505, 450411118044836946050)
(1365200765625, 546409811106791250)
(-1101502139991, 689634668272172502)
(-3413830263855/4, 7076061679895292975/8)
(5741529645105/4, -6176268700616275425/8)
(2531821884780, 3473362136846698350)

Commentary

Quartic twist y^2 = x^3 + a x with a = -1645077324548360946504525. 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 21:29:26 UTC · history