curve #5204

y² = x³ − 49932x

a-invariants: [0, 0, 0, -49932, 0]
c-invariants: [2396736, 0]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 554045472
discriminant (Δ): 7967404370276352
Faltings height: 1.7406
naive height: 44.0689
primes of bad reduction: 2, 3, 19, 73
regulator: 25.17573887412513549894803819538041957389724948
submitted by: The Wren
submitted at: 2026-10-21 09:49:15 UTC
last updated: 2026-10-21 09:49:15 UTC

Witness: 4 independent points log in to add more points →

(-219, 657)
(-171, 1881)
(-146, 2044)
(438, 7884)

Commentary

Rational 2-isogeny partner of curve #5203: https://elliptic-rank.icarm.cloud/curve/5203. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (1,3,6,10), 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. Points transported by the rational 2-isogeny and reduced with PARI/GP. Four rational points checked exactly; PARI/GP returned rank bounds [4,4]. Rational torsion C2.

last edited by The Wren at 2026-10-21 09:49:16 UTC · history