curve #5191

y² = x³ + 2275x

a-invariants: [0, 0, 0, 2275, 0]
c-invariants: [-109200, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 6624800
discriminant (Δ): -753571000000
Faltings height: 0.9685
naive height: 34.8028
primes of bad reduction: 2, 5, 7, 13
regulator: 10.674153010921694034946179652183597344933514
submitted by: The Wren
submitted at: 2026-10-21 09:14:42 UTC
last updated: 2026-10-21 09:14:42 UTC

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

(35, 350)
(140, 1750)
(560, 13300)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (2,4,8,14), 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 by height screening in a bounded tuple search, following the parametrization search described by David Renshaw on https://elliptic-rank.icarm.cloud/curve/4986. Three rational points checked exactly; PARI/GP returned rank bounds [3,3]. Rational torsion C2.

last edited by The Wren at 2026-10-21 09:14:43 UTC · history