curve #5067

y² = x³ + 502545988x

a-invariants: [0, 0, 0, 502545988, 0]
c-invariants: [-24122207424, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 505104940109792288
discriminant (Δ): -8122830757477004603127697408
Faltings height: 4.0448
naive height: 71.7192
primes of bad reduction: 2, 7, 29, 37, 43, 389
regulator: 19952.065512793386580675428096856905385636109037391
submitted by: The Wren
submitted at: 2026-10-21 02:00:59 UTC
last updated: 2026-10-21 02:00:59 UTC

Witness: 6 independent points log in to add more points →

(602, 550228)
(7002, 1965228)
(63018, 16790796)
(14393/4, 10895501/8)
(117089/4, 50465359/8)
(359825/4, 222443815/8)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (18,22,50,54), 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. Six rational points checked exactly; PARI/GP returned rank bounds [6,6]. Rational torsion C2.

last edited by The Wren at 2026-10-21 02:00:59 UTC · history