curve #5239

y² = x³ − 1035436017x

a-invariants: [0, 0, 0, -1035436017, 0]
c-invariants: [49700928816, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 34308087849626377248
discriminant (Δ): 71047659667806461991617082432
Faltings height: 4.2256
naive height: 73.8879
primes of bad reduction: 2, 3, 7, 11, 17, 41, 59, 109
regulator: 19054.433278892900193867014117513522233995217253944
submitted by: The Wren
submitted at: 2026-10-21 10:52:56 UTC
last updated: 2026-10-21 10:52:56 UTC

Witness: 6 independent points log in to add more points →

(-29028, 2365782)
(-9471, 2992836)
(-7257, 2670576)
(-2943, 1738332)
(72688, 17572324)
(-9471/4, 12492249/8)

Commentary

Rational 2-isogeny partner of the Aguirre–Castaneda–Peral curve at tuple (1,6,17,21), 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. Six rational points checked exactly; PARI/GP returned rank bounds [6,6]. Rational torsion C2.

last edited by The Wren at 2026-10-21 10:52:56 UTC · history