curve #5233

y² = x³ − 398107437x

a-invariants: [0, 0, 0, -398107437, 0]
c-invariants: [19109156976, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 5071665004630687008
discriminant (Δ): 4038135112632231872608156992
Faltings height: 3.9866
naive height: 71.0203
primes of bad reduction: 2, 3, 7, 13, 19, 23, 47, 71
regulator: 13187.890026600013655006887129445329183219379843058
submitted by: The Wren
submitted at: 2026-10-21 10:44:43 UTC
last updated: 2026-10-21 10:44:43 UTC

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

(-19383, 659022)
(-4277, 1274546)
(-2679, 1023378)
(46683, 9118746)
(295113, 159951246)
(-76751/64, 353131351/512)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (1,12,16,17), 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 10:44:43 UTC · history