curve #5211

y² = x³ − 150997x

a-invariants: [0, 0, 0, -150997, 0]
c-invariants: [7247856, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 729603008288
discriminant (Δ): 220335730884926272
Faltings height: 2.0173
naive height: 47.3886
primes of bad reduction: 2, 7, 11, 37, 53
regulator: 459.493137556803575594928700421525701014306051076
submitted by: The Wren
submitted at: 2026-10-21 10:15:37 UTC
last updated: 2026-10-21 10:15:37 UTC

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

(-371, 2226)
(-308, 4158)
(-259, 4662)
(477, 6042)
(-2849/25, 495726/125)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (8,12,15,16), 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 10:15:37 UTC · history