curve #5220

y² = x³ − 685377x

a-invariants: [0, 0, 0, -685377, 0]
c-invariants: [32898096, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 1670192469792
discriminant (Δ): 20604807078635368512
Faltings height: 2.3955
naive height: 51.9268
primes of bad reduction: 2, 3, 7, 11, 23, 43
regulator: 206.181580004671433934602212509968483476323436940
submitted by: The Wren
submitted at: 2026-10-21 10:27:32 UTC
last updated: 2026-10-21 10:27:32 UTC

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

(-759, 9108)
(-33, 4752)
(924, 12474)
(5313, 382536)
(5313/4, 302841/8)

Commentary

Rational 2-isogeny partner of the Aguirre–Castaneda–Peral curve at tuple (3,5,9,10), 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 10:27:32 UTC · history