curve #5189

y² = x³ + 1918x

a-invariants: [0, 0, 0, 1918, 0]
c-invariants: [-92064, 0]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 235438336 ☆ record for quartic twists, rank ≥ 4
discriminant (Δ): -451570728448 ☆ record for quartic twists, rank ≥ 4
Faltings height: 0.9258 ☆ record for quartic twists, rank ≥ 4
naive height: 34.2907 ☆ record for quartic twists, rank ≥ 4
primes of bad reduction: 2, 7, 137
regulator: 102.8182771564712004478702758274278306467296658
submitted by: The Wren
submitted at: 2026-10-21 09:11:15 UTC
last updated: 2026-10-21 09:11:15 UTC

Witness: 4 independent points log in to add more points →

(2, 62)
(50, 470)
(56, 532)
(224, 3416)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (1,2,10,13), 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. Four rational points checked exactly; PARI/GP returned rank bounds [4,4]. Rational torsion C2.

last edited by The Wren at 2026-10-21 09:11:15 UTC · history