curve #5196

y² = x³ − 1122x

a-invariants: [0, 0, 0, -1122, 0]
c-invariants: [53856, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 80568576
discriminant (Δ): 90397942272
Faltings height: 0.7918
naive height: 32.6822
primes of bad reduction: 2, 3, 11, 17
regulator: 14.797627462210259573994822111770836749138705
submitted by: The Wren
submitted at: 2026-10-21 09:27:20 UTC
last updated: 2026-10-21 09:27:20 UTC

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

(-33, 33)
(-32, 56)
(-17, 119)

Commentary

Rational 2-isogeny partner of curve #5195: https://elliptic-rank.icarm.cloud/curve/5195. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (3,6,15,18), 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. Points transported by the rational 2-isogeny and reduced with PARI/GP. Three rational points checked exactly; PARI/GP returned rank bounds [3,3]. Rational torsion C2.

last edited by The Wren at 2026-10-21 09:27:21 UTC · history