curve #5237

y² = x³ − 2172367197x

a-invariants: [0, 0, 0, -2172367197, 0]
c-invariants: [104273625456, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 16779303959472486432 ☆ record for quartic twists, rank ≥ 7
discriminant (Δ): 656114571180904445080278679872
Faltings height: 4.4108
naive height: 76.1109
primes of bad reduction: 2, 3, 7, 11, 13, 59, 61, 67
regulator: 26578.18355823189941634549594659745069911131874740839
submitted by: The Wren
submitted at: 2026-10-21 10:50:24 UTC
last updated: 2026-10-21 10:50:24 UTC

Witness: 7 independent points log in to add more points →

(-25311, 6226506)
(-9581, 4464746)
(48321, 2802618)
(343308, 199290294)
(-327509/9, 149999122/27)
(-154167/4, 41162589/8)
(-118767/4, 49525839/8)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (10,13,18,19), 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. Seven rational points checked exactly; PARI/GP returned rank bounds [7,7]. Rational torsion C2.

last edited by The Wren at 2026-10-21 10:50:24 UTC · history