curve #5225

y² = x³ + 4664975x

a-invariants: [0, 0, 0, 4664975, 0]
c-invariants: [-223918800, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 27855349440800
discriminant (Δ): -6497225437879799000000
Faltings height: 2.8749
naive height: 57.6804
primes of bad reduction: 2, 5, 7, 19, 23, 61
regulator: 898.631325258750830562534804877865895171342937948
submitted by: The Wren
submitted at: 2026-10-21 10:32:36 UTC
last updated: 2026-10-21 10:32:36 UTC

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

(380, 42750)
(1159, 83448)
(2745, 183000)
(11495, 1254000)
(305/4, 150975/8)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (5,10,12,15), 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 10:32:36 UTC · history