curve #5247

y² = x³ + 5643558023103x

a-invariants: [0, 0, 0, 5643558023103, 0]
c-invariants: [-270890785108944, 0]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 113243545458240858601772832
discriminant (Δ): -11503737351638720895791563856874247278528
Faltings height: 6.3764
naive height: 99.6982
primes of bad reduction: 2, 3, 7, 11, 13, 17, 19, 23, 37, 43, 53
regulator: 16211921.6160584152378015493148246600713726769035597296
submitted by: The Wren
submitted at: 2026-10-21 11:03:33 UTC
last updated: 2026-10-21 11:03:33 UTC

Witness: 8 independent points log in to add more points →

(757575, 2170300860)
(2026332, 4444759242)
(3867721, 8926700068)
(6084351, 16111361448)
(53535559/25, 585036588752/125)
(108971632/9, 1159237744588/27)
(481820031/259081, 13509977340983508/131872229)
(976200316/729, 62113802153858/19683)

Commentary

Rational 2-isogeny partner of the Aguirre–Castaneda–Peral curve at tuple (2,14,18,22), 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. Eight rational points checked exactly; PARI/GP returned rank bounds [8,8]. Rational torsion C2.

last edited by The Wren at 2026-10-21 11:03:33 UTC · history