curve #5349

y² = x³ + 257381680598x

a-invariants: [0, 0, 0, 257381680598, 0]
c-invariants: [-12354320668704, 0]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 4239701088476856936806656
discriminant (Δ): -1091221391385343330424111404672460288
Faltings height: 5.6045
naive height: 90.4351
primes of bad reduction: 2, 7, 19, 31, 109, 233, 1229
regulator: 3855956.16960795720995100782655706858493114523231461077
submitted by: The Wren
submitted at: 2026-10-21 15:27:08 UTC
last updated: 2026-10-21 15:27:08 UTC

Witness: 8 independent points log in to add more points →

(2458, 25152714)
(15463, 63115689)
(158922, 211937202)
(296856, 320258148)
(449407, 454350477)
(937727, 1032437427)
(168664273/36, 2203261398199/216)
(168664273/1024, 7093513329561/32768)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (3,8,16,30), 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. Eight supplied rational points checked exactly; PARI/GP returned rank bounds [8,8]. Rational torsion C2.

last edited by The Wren at 2026-10-21 15:27:09 UTC · history