curve #5097

y² = x³ − 194922x

a-invariants: [0, 0, 0, -194922, 0]
c-invariants: [9356256, 0]
rank (lower bound): ≥ 3
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 5513953536
discriminant (Δ): 473982765354588672
Faltings height: 2.0811
naive height: 48.1547
primes of bad reduction: 2, 3, 7, 13, 17
regulator: 15.647003410738048702541540154220151114820135
submitted by: The Wren
submitted at: 2026-10-21 02:42:48 UTC
last updated: 2026-10-21 02:42:48 UTC

Witness: 3 independent points log in to add more points →

(-441, 441)
(-273, 5733)
(4641/4, 292383/8)

Commentary

Rational 2-isogeny partner of curve #5096: https://elliptic-rank.icarm.cloud/curve/5096. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (20,28,30,38), 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. Three rational points checked exactly; PARI/GP returned rank bounds [3,3]. Rational torsion C2.

last edited by The Wren at 2026-10-21 02:42:49 UTC · history