curve #5229

y² = x³ − 12743577x

a-invariants: [0, 0, 0, -12743577, 0]
c-invariants: [611691696, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 11784036626208
discriminant (Δ): 132450626299107445826112
Faltings height: 3.1262
naive height: 60.6952
primes of bad reduction: 2, 3, 7, 11, 37, 71
regulator: 289.471490977468886254403248187119667462581338913
submitted by: The Wren
submitted at: 2026-10-21 10:38:53 UTC
last updated: 2026-10-21 10:38:53 UTC

Witness: 5 independent points log in to add more points →

(-3108, 97902)
(-2331, 130536)
(-1221, 117216)
(8547, 717948)
(25641/4, 3410253/8)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (4,5,14,20), 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. Selected in a bounded tuple search following the parametrization search described by David Renshaw on https://elliptic-rank.icarm.cloud/curve/4986. Five rational points checked exactly; PARI/GP returned rank bounds [5,5]. Rational torsion C2.

last edited by The Wren at 2026-10-21 10:38:53 UTC · history