curve #5213

y² = x³ − 49042x

a-invariants: [0, 0, 0, -49042, 0]
c-invariants: [2354016, 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): 7548914264453632 ☆ record for quartic twists, rank ≥ 5
Faltings height: 1.7361 ☆ record for quartic twists, rank ≥ 5
naive height: 44.0149 ☆ record for quartic twists, rank ≥ 5
primes of bad reduction: 2, 7, 31, 113
regulator: 530.109354889906775400389287107946713916703464642
submitted by: The Wren
submitted at: 2026-10-21 10:19:20 UTC
last updated: 2026-10-21 10:19:20 UTC

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

(-217, 651)
(-62, 1674)
(224, 504)
(343, 4851)
(-1736/9, 40796/27)

Commentary

Rational 2-isogeny partner of the Aguirre–Castaneda–Peral curve 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. Selected in a bounded tuple search following the parametrization search 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 10:19:20 UTC · history