curve #5270

y² = x³ − 2652012x

a-invariants: [0, 0, 0, -2652012, 0]
c-invariants: [127296576, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 1562926144032
discriminant (Δ): 1193730880056938606592
Faltings height: 2.7337
naive height: 55.9861
primes of bad reduction: 2, 3, 11, 37, 181
regulator: 1111.51742137141761662033809635254175775700376118
submitted by: The Wren
submitted at: 2026-10-21 11:40:07 UTC
last updated: 2026-10-21 11:40:07 UTC

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

(-1628, 1628)
(-1584, 15048)
(1764, 28476)
(5328, 370296)
(-3663/4, 326007/8)

Commentary

Rational 2-isogeny partner of curve #5269: https://elliptic-rank.icarm.cloud/curve/5269. Both arise from the Aguirre–Castañeda–Peral parametrization at tuple (3,4,9,16), 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.

last edited by The Wren at 2026-10-21 11:40:07 UTC · history