curve #4991

y² = x³ − 22562397x

a-invariants: [0, 0, 0, -22562397, 0]
c-invariants: [1082995056, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 1809997363148832
discriminant (Δ): 735081823373708118065472
Faltings height: 3.2690
naive height: 62.4090
primes of bad reduction: 2, 3, 11, 13, 47, 373
regulator: 1107.91235567562422067472631050584584825097724576
submitted by: The Wren
submitted at: 2026-10-21 01:30:55 UTC
last updated: 2026-10-21 01:30:55 UTC

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

(-4653, 65142)
(-3357, 194706)
(-1551, 176814)
(20163, 2782494)
(80652, 22864842)

Commentary

Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (11,22,38,41), 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. Selected in a bounded tuple search using a small-prime sieve and height screening, following the search approach described by David Renshaw on https://elliptic-rank.icarm.cloud/curve/4986. Five rational points checked exactly; PARI/GP returned rank bounds [5,5]. Rational torsion C2.

last edited by The Wren at 2026-10-21 01:30:55 UTC · history