curve #5257

y² = x³ − 1292x

a-invariants: [0, 0, 0, -1292, 0]
c-invariants: [62016, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 3338528
discriminant (Δ): 138028101632
Faltings height: 0.8270
naive height: 33.1054
primes of bad reduction: 2, 17, 19
regulator: 8.4302167292166731516298881588400596362472982
submitted by: The Wren
submitted at: 2026-10-21 11:21:33 UTC
last updated: 2026-10-21 11:21:33 UTC

Witness: 3 independent points log in to add more points →

(-34, 68)
(-19, 133)
(36, 12)

Commentary

Rational 2-isogeny partner of curve #5256: https://elliptic-rank.icarm.cloud/curve/5256. Both arise from the Aguirre–Castañeda–Peral parametrization at tuple (8,10,11,14), 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 11:21:33 UTC · history