curve #5214

y² = x³ + 196168x

a-invariants: [0, 0, 0, 196168, 0]
c-invariants: [-9416064, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 153927536896 ☆ record for quartic twists, rank ≥ 5
discriminant (Δ): -483130512925032448
Faltings height: 2.0827
naive height: 48.1738
primes of bad reduction: 2, 7, 31, 113
regulator: 1060.21870977981355080077857421589342783340692928
submitted by: The Wren
submitted at: 2026-10-21 10:20:34 UTC
last updated: 2026-10-21 10:20:34 UTC

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

(32, 2512)
(113, 4859)
(128, 5216)
(2825, 151985)
(113/4, 18871/8)

Commentary

Rational 2-isogeny partner of curve #5213: https://elliptic-rank.icarm.cloud/curve/5213. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (1,2,10,11), 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 10:20:34 UTC · history