curve #5314

y² = x³ − 101016972x

a-invariants: [0, 0, 0, -101016972, 0]
c-invariants: [4848814656, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 2267650807121952
discriminant (Δ): 65972510809578900221571072
Faltings height: 3.6437
naive height: 66.9060
primes of bad reduction: 2, 3, 7, 31, 67, 193
regulator: 2499.2470058419056743172315475988093254614120241403
submitted by: The Wren
submitted at: 2026-10-21 13:04:19 UTC
last updated: 2026-10-21 13:04:19 UTC

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

(-10050, 12060)
(-8106, 534996)
(-5628, 624708)
(10854, 426924)
(24318, 3453156)
(-77586/25, 66568788/125)

Commentary

Rational 2-isogeny partner of curve #5312: https://elliptic-rank.icarm.cloud/curve/5312. Both arise from the Aguirre–Castañeda–Peral parametrization at tuple (6,11,18,22), 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 13:04:19 UTC · history