curve #5294

y² = x³ + 10617420723x

a-invariants: [0, 0, 0, 10617420723, 0]
c-invariants: [-509636194704, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 3607347929894074967328
discriminant (Δ): -76601461331857005506047710276288
Faltings height: 4.8075
naive height: 80.8709
primes of bad reduction: 2, 3, 7, 13, 19, 43, 181, 263
regulator: 180681.0880867547145922203030008481433848128592064263
submitted by: The Wren
submitted at: 2026-10-21 12:20:32 UTC
last updated: 2026-10-21 12:20:32 UTC

Witness: 7 independent points log in to add more points →

(24073, 16417786)
(58825, 28777190)
(74347, 34645702)
(101179, 45935266)
(284713, 161561686)
(1035139/9, 1413999874/27)
(4140556/25, 9922842454/125)

Commentary

Rational 2-isogenous endpoint of the Aguirre–Castañeda–Peral parametrization at tuple (10,12,16,17), 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.

last edited by The Wren at 2026-10-21 12:20:32 UTC · history