curve #5093

y² = x³ − 13617x

a-invariants: [0, 0, 0, -13617, 0]
c-invariants: [653616, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 659280672
discriminant (Δ): 161593648391232
Faltings height: 1.4158
naive height: 40.1708
primes of bad reduction: 2, 3, 17, 89
regulator: 19.371393622973145695542936912196162683409333
submitted by: The Wren
submitted at: 2026-10-21 02:37:40 UTC
last updated: 2026-10-21 02:37:40 UTC

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

(-89, 712)
(-81, 756)
(801, 22428)

Commentary

Rational 2-isogeny partner of curve #5092: https://elliptic-rank.icarm.cloud/curve/5092. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (3,9,33,45), 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 02:37:40 UTC · history