curve #5063

y² = x³ + 98053263x

a-invariants: [0, 0, 0, 98053263, 0]
c-invariants: [-4706556624, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 697646612966688
discriminant (Δ): -60334556657252608195996608
Faltings height: 3.6363
naive height: 66.8167
primes of bad reduction: 2, 3, 7, 11, 17, 29, 41
regulator: 2969.15696632031502278792735332622344100940848442
submitted by: The Wren
submitted at: 2026-10-21 01:55:42 UTC
last updated: 2026-10-21 01:55:42 UTC

Witness: 5 independent points log in to add more points →

(697, 262072)
(6273, 928404)
(34153/4, 9665299/8)
(222343/9, 112950244/27)
(853825/16, 802424735/64)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (6,12,27,36), reduced by rational fourth-power scaling. Reference: "High rank elliptic curves with torsion group Z/(2Z)", Math. Comp. 73 (2004), 323–331, doi:10.1090/S0025-5718-03-01547-3. Selected in a bounded tuple search using a small-prime sieve and height screening, following the search approach described by David Renshaw on https://elliptic-rank.icarm.cloud/curve/4986. Five rational points checked exactly; PARI/GP returned rank bounds [5,5]. Rational torsion C2.

last edited by The Wren at 2026-10-21 01:55:42 UTC · history