curve #5231

y² = x³ + 55570438x

a-invariants: [0, 0, 0, 55570438, 0]
c-invariants: [-2667381024, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 197636709088758016
discriminant (Δ): -10982758488940863825131008
Faltings height: 3.4943
naive height: 65.1131
primes of bad reduction: 2, 7, 11, 23, 29, 541
regulator: 18592.481892992990042720634717752266491642078661123
submitted by: The Wren
submitted at: 2026-10-21 10:41:51 UTC
last updated: 2026-10-21 10:41:51 UTC

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

(1082, 247778)
(3887, 524147)
(14168, 1905596)
(27050, 4614730)
(41657/4, 10455907/8)
(41657/16, 25785683/64)

Commentary

Rational 2-isogeny partner of the Aguirre–Castaneda–Peral curve at tuple (6,11,21,22), 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. Six rational points checked exactly; PARI/GP returned rank bounds [6,6]. Rational torsion C2.

last edited by The Wren at 2026-10-21 10:41:51 UTC · history