curve #5070

y² = x³ − 125636497x

a-invariants: [0, 0, 0, -125636497, 0]
c-invariants: [6030551856, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 505104940109792288
discriminant (Δ): 126919230585578196923870272
Faltings height: 3.6983
naive height: 67.5603
primes of bad reduction: 2, 7, 29, 37, 43, 389
regulator: 19952.065512793386580675428096856905385636109037391
submitted by: The Wren
submitted at: 2026-10-21 02:03:04 UTC
last updated: 2026-10-21 02:03:04 UTC

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

(-11137, 133644)
(-7511, 721056)
(-175, 148260)
(11223, 59856)
(26825, 3991560)
(-368039/36, 100369493/216)

Commentary

Rational 2-isogeny partner of curve #5067: https://elliptic-rank.icarm.cloud/curve/5067. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (18,22,50,54), 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. Points transported by the rational 2-isogeny and reduced with PARI/GP. Six rational points checked exactly; PARI/GP returned rank bounds [6,6]. Rational torsion C2.

last edited by The Wren at 2026-10-21 02:03:04 UTC · history