curve #5250

y² = x³ + 4034093166186533953503x

a-invariants: [0, 0, 0, 4034093166186533953503, 0]
c-invariants: [-193636471976953629768144, 0]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 10627858072472094204174121166893361218593312
discriminant (Δ): -4201629422893286305152814322400756714397872868867397507408127649728
Faltings height: 11.4733
naive height: 160.8608
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 11:09:17 UTC
last updated: 2026-10-21 11:09:17 UTC

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

(1916259513, 2781619313956644)
(4593622033, 4316020266301744)
(7486596348, 5533646567958894)
(30295150396, 12248192976926018)
(138613224633, 56766551110401756)
(151603576047, 63999887594127408)
(993247230976, 991909792540514848)
(520244904111/20449, 936804545179121315208/2924207)
(512427725025216/7921, 16255890186395368701192/704969)
(2594165357940156/37249, 178824344135504418729378/7189057)

Commentary

Rational 2-isogeny partner of curve #5249: https://elliptic-rank.icarm.cloud/curve/5249. 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. Ten rational points checked exactly; PARI/GP returned rank bounds [10,10]. Rational torsion C2.

last edited by The Wren at 2026-10-21 11:09:17 UTC · history