curve #5217

y² = x³ − 7262332x

a-invariants: [0, 0, 0, -7262332, 0]
c-invariants: [348591936, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 105482932156448
discriminant (Δ): 24513666356915242135552
Faltings height: 2.9856
naive height: 59.0082
primes of bad reduction: 2, 7, 11, 17, 19, 73
regulator: 640.033719836462018120949235638353914690423475154
submitted by: The Wren
submitted at: 2026-10-21 10:23:30 UTC
last updated: 2026-10-21 10:23:30 UTC

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

(-2142, 75684)
(-1606, 86724)
(-1022, 79716)
(-374, 51612)
(30514, 5309436)

Commentary

Rational 2-isogeny partner of curve #5216: https://elliptic-rank.icarm.cloud/curve/5216. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (4,7,12,14), 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. Five rational points checked exactly; PARI/GP returned rank bounds [5,5]. Rational torsion C2.

last edited by The Wren at 2026-10-21 10:23:30 UTC · history