curve #5218

y² = x³ + 613263x

a-invariants: [0, 0, 0, 613263, 0]
c-invariants: [-29436624, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 12034928229408
discriminant (Δ): -14761152381502876608
Faltings height: 2.3677
naive height: 51.5933
primes of bad reduction: 2, 3, 7, 19, 29, 53
regulator: 820.732876610346838150920922371160712295826185983
submitted by: The Wren
submitted at: 2026-10-21 10:24:55 UTC
last updated: 2026-10-21 10:24:55 UTC

Witness: 5 independent points log in to add more points →

(57, 5928)
(399, 17556)
(513, 21204)
(1596, 71022)
(3591, 220248)

Commentary

Rational 2-isogeny partner of the Aguirre–Castaneda–Peral curve at tuple (1,2,9,22), 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. Five rational points checked exactly; PARI/GP returned rank bounds [5,5]. Rational torsion C2.

last edited by The Wren at 2026-10-21 10:24:55 UTC · history