curve #5194

y² = x³ − 1045x

a-invariants: [0, 0, 0, -1045, 0]
c-invariants: [50160, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 34944800
discriminant (Δ): 73034632000
Faltings height: 0.7740
naive height: 32.4689
primes of bad reduction: 2, 5, 11, 19
regulator: 16.727772930469645030824853662433559580558195
submitted by: The Wren
submitted at: 2026-10-21 09:24:08 UTC
last updated: 2026-10-21 09:24:08 UTC

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

(-19, 114)
(44, 198)
(125, 1350)

Commentary

Rational 2-isogeny partner of curve #5193: https://elliptic-rank.icarm.cloud/curve/5193. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (1,4,8,13), 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 09:24:08 UTC · history