curve #5249

y² = x³ − 16136372664746135814012x

a-invariants: [0, 0, 0, -16136372664746135814012, 0]
c-invariants: [774545887907814519072576, 0]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 10627858072472094204174121166893361218593312
discriminant (Δ): 268904283065170323529780116633648429721463863607513440474120169582592
Faltings height: 11.8199
naive height: 165.0197
primes of bad reduction: 2, 3, 7, 11, 13, 23, 31, 47, 53, 67, 83, 199, 251, 389
regulator: 41624158494.12960260592823494872155269337593272704107183244
submitted by: The Wren
submitted at: 2026-10-21 11:08:03 UTC
last updated: 2026-10-21 11:08:03 UTC

Witness: 10 independent points log in to add more points →

(-126737391326, 3062575058678044)
(-116520927302, 17268634468011004)
(-115145074316, 18204003462700964)
(422968268514, 262382853141031524)
(374305734783253, 7241686111799127886079)
(-9506280126251/81, 12139211389446060191/729)
(-2464330016876/81, 15682307229718263316/729)
(5229225804594/25, 9500220351566871828/125)
(30860691554368/49, 167915237001935313904/343)
(157343106240853/841, 1448989714430967781631/24389)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (5,14,19,21), 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. Ten rational points checked exactly; PARI/GP returned rank bounds [10,10]. Rational torsion C2.

last edited by The Wren at 2026-10-21 11:08:03 UTC · history