curve #4994

y² = x³ + 315672308x

a-invariants: [0, 0, 0, 315672308, 0]
c-invariants: [-15152270784, 0]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 4067306368899872
discriminant (Δ): -2013211630459916119746695168
Faltings height: 3.9286
naive height: 70.3242
primes of bad reduction: 2, 7, 19, 29, 37, 79
regulator: 1234.12170754437686570019756369074145476388116797
submitted by: The Wren
submitted at: 2026-10-21 01:35:26 UTC
last updated: 2026-10-21 01:35:26 UTC

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

(406, 358092)
(1862, 770868)
(12950, 2501940)
(19684, 3720276)
(20461/9, 23059547/27)

Commentary

Rational 2-isogeny partner of curve #4993: https://elliptic-rank.icarm.cloud/curve/4993. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (14,28,32,34), 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. Five rational points checked exactly; PARI/GP returned rank bounds [5,5]. Rational torsion C2.

last edited by The Wren at 2026-10-21 01:35:26 UTC · history