curve #5066

y² = x³ − 31131177x

a-invariants: [0, 0, 0, -31131177, 0]
c-invariants: [1494296496, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 31012805804970528
discriminant (Δ): 1930930293562329973902912
Faltings height: 3.3495
naive height: 63.3748
primes of bad reduction: 2, 3, 7, 11, 19, 41, 173
regulator: 1467.57362871770068043300496144600775241768999177
submitted by: The Wren
submitted at: 2026-10-21 01:59:38 UTC
last updated: 2026-10-21 01:59:38 UTC

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

(-5453, 87248)
(-4389, 228228)
(-4325, 231820)
(6517, 271852)
(179949, 76298376)

Commentary

Rational 2-isogeny partner of curve #5065: https://elliptic-rank.icarm.cloud/curve/5065. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (1,7,10,14), 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. Points transported by the rational 2-isogeny and reduced with PARI/GP. Five rational points checked exactly; PARI/GP returned rank bounds [5,5]. Rational torsion C2.

last edited by The Wren at 2026-10-21 01:59:38 UTC · history