curve #5219

y² = x³ − 2453052x

a-invariants: [0, 0, 0, -2453052, 0]
c-invariants: [117746496, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 12034928229408
discriminant (Δ): 944713752416184102912
Faltings height: 2.7143
naive height: 55.7521
primes of bad reduction: 2, 3, 7, 19, 29, 53
regulator: 410.366438305173419075460461185580356147913092998
submitted by: The Wren
submitted at: 2026-10-21 10:26:05 UTC
last updated: 2026-10-21 10:26:05 UTC

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

(-1484, 19292)
(-812, 38164)
(-318, 27348)
(6612, 522348)
(14098, 1663564)

Commentary

Rational 2-isogeny partner of curve #5218: https://elliptic-rank.icarm.cloud/curve/5218. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (1,2,9,22), 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 10:26:05 UTC · history