curve #4768

y² = x³ + 1292292705051108627666723x

a-invariants: [0, 0, 0, 1292292705051108627666723, 0]
c-invariants: [-62030049842453214128002704, 0]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 5937850437433996936516077156029887017540129986592
discriminant (Δ): -138121934471648793849592387218589548500103800986006285218689765056222404288
Faltings height: 12.9157
naive height: 178.1690
primes of bad reduction: 2, 3, 7, 11, 19, 23, 47, 67, 103, 127, 149, 353, 733, 2687
regulator: 1963671448056.6627503530283786458452920277535424219535508229517
submitted by: David Renshaw
submitted at: 2026-10-20 21:29:22 UTC
last updated: 2026-10-20 21:29:22 UTC

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

(2577978443601, -4523790169219049982)
(1431726267987, -2187468705449009898)
(327853720593, -677438803529294814)
(1517017917825, 2334868370032644090)
(1278942731208307/289, -47222880788985791267818/4913)
(78260155090425/169, -1835165858057544065130/2197)
(2496275984304, 4333727946071531484)
(10230307993675/9, -46276998723425538530/27)
(1278942731208307/5041, 209979956135242228445378/357911)
(6742301857, 93345240911833898)
(2942554657497904/625, -164206860058650562726292/15625)
(5750386997425/16, -45744187104995738345/64)

Commentary

Quartic twist y^2 = x^3 + a x with a = 1292292705051108627666723 (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 21:29:22 UTC · history