curve #5232

y² = x³ − 222281752x

a-invariants: [0, 0, 0, -222281752, 0]
c-invariants: [10669524096, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 197636709088758016
discriminant (Δ): 702896543292215284808384512
Faltings height: 3.8409
naive height: 69.2720
primes of bad reduction: 2, 7, 11, 23, 29, 541
regulator: 167332.33703693691038448571245977039842477870795011
submitted by: The Wren
submitted at: 2026-10-21 10:43:32 UTC
last updated: 2026-10-21 10:43:32 UTC

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

(-6496, 1081584)
(-3087, 810411)
(-2023, 664377)
(-8679671/2209, 93598233729/103823)
(-22743/4, 8313963/8)
(-1624/25, 15020376/125)

Commentary

Rational 2-isogeny partner of curve #5231: https://elliptic-rank.icarm.cloud/curve/5231. Obtained from the Aguirre–Castaneda–Peral parametrization 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. Six rational points checked exactly; PARI/GP returned rank bounds [6,6]. Rational torsion C2.

last edited by The Wren at 2026-10-21 10:43:32 UTC · history