curve #5064

y² = x³ − 392213052x

a-invariants: [0, 0, 0, -392213052, 0]
c-invariants: [18826226496, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 697646612966688
discriminant (Δ): 3861411626064166924543782912
Faltings height: 3.9829
naive height: 70.9756
primes of bad reduction: 2, 3, 7, 11, 17, 29, 41
regulator: 371.144620790039377848490919165777930126176060557
submitted by: The Wren
submitted at: 2026-10-21 01:57:02 UTC
last updated: 2026-10-21 01:57:02 UTC

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

(-13398, 1688148)
(-5166, 1374156)
(-3654, 1176588)
(20706, 869652)
(183106/9, 17944388/27)

Commentary

Rational 2-isogeny partner of curve #5063: https://elliptic-rank.icarm.cloud/curve/5063. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (6,12,27,36), 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 01:57:02 UTC · history