curve #5198

y² = x³ + 20148x

a-invariants: [0, 0, 0, 20148, 0]
c-invariants: [-967104, 0]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 811883808
discriminant (Δ): -523450718834688
Faltings height: 1.5138
naive height: 41.3462
primes of bad reduction: 2, 3, 23, 73
regulator: 64.00045205127535212964565537438378414468288753
submitted by: The Wren
submitted at: 2026-10-21 09:30:27 UTC
last updated: 2026-10-21 09:30:27 UTC

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

(4, 284)
(6, 348)
(46, 1012)
(69, 1311)

Commentary

Rational 2-isogeny partner of curve #5197: https://elliptic-rank.icarm.cloud/curve/5197. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (1,12,20,21), 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 09:30:27 UTC · history