curve #5245

y² = x³ + 111372406068x

a-invariants: [0, 0, 0, 111372406068, 0]
c-invariants: [-5345875491264, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 205021699725214598688
discriminant (Δ): -88412318698890492183235619862171648
Faltings height: 5.3951
naive height: 87.9220
primes of bad reduction: 2, 3, 7, 11, 17, 41, 79, 199
regulator: 147406.9949577457017749758019091921027028725250210090
submitted by: The Wren
submitted at: 2026-10-21 11:00:21 UTC
last updated: 2026-10-21 11:00:21 UTC

Witness: 7 independent points log in to add more points →

(29766, 57805572)
(88806, 102912132)
(111078, 117224316)
(337348, 275613316)
(407950, 336640340)
(1974478/9, 5050714724/27)
(2108425/4, 3623539205/8)

Commentary

Rational 2-isogeny partner of curve #5244: https://elliptic-rank.icarm.cloud/curve/5244. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (5,8,11,18), 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. Seven rational points checked exactly; PARI/GP returned rank bounds [7,7]. Rational torsion C2.

last edited by The Wren at 2026-10-21 11:00:21 UTC · history