curve #5227

y² = x³ − 2192337x

a-invariants: [0, 0, 0, -2192337, 0]
c-invariants: [105232176, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 17089214298912
discriminant (Δ): 674375702551809072192
Faltings height: 2.6862
naive height: 55.4150
primes of bad reduction: 2, 3, 7, 17, 23, 89
regulator: 858.867021545397571729520774073057331733652909687
submitted by: The Wren
submitted at: 2026-10-21 10:36:12 UTC
last updated: 2026-10-21 10:36:12 UTC

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

(-1449, 11592)
(-1377, 20196)
(-801, 35244)
(-425, 29240)
(13617, 1579572)

Commentary

Rational 2-isogeny partner of the Aguirre–Castaneda–Peral curve at tuple (3,7,15,19), 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 10:36:12 UTC · history