curve #5246

y² = x³ + 39736873638x

a-invariants: [0, 0, 0, 39736873638, 0]
c-invariants: [-1907369934624, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 11228580455270253191424
discriminant (Δ): -4015698144166715059870935419524608
Faltings height: 5.1374
naive height: 84.8302
primes of bad reduction: 2, 3, 7, 11, 19, 29, 61, 853
regulator: 144011.4618306429118136304837208831535586683307290336
submitted by: The Wren
submitted at: 2026-10-21 11:01:29 UTC
last updated: 2026-10-21 11:01:29 UTC

Witness: 7 independent points log in to add more points →

(5016, 14122548)
(194712, 122960628)
(268888, 173567204)
(705831, 616190463)
(2451834, 3851831214)
(58377/4, 193169493/8)
(705831/25, 4228633521/125)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (4,8,17,22), 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. Seven rational points checked exactly; PARI/GP returned rank bounds [7,7]. Rational torsion C2.

last edited by The Wren at 2026-10-21 11:01:29 UTC · history