curve #5075

y² = x³ + 42966979188x

a-invariants: [0, 0, 0, 42966979188, 0]
c-invariants: [-2062415001024, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 3692322601084050278688
discriminant (Δ): -5076734347397133266020252670251008
Faltings height: 5.1570
naive height: 85.0647
primes of bad reduction: 2, 3, 7, 17, 29, 61, 73, 233
regulator: 43689.570846531524737657421443965615487277163118130
submitted by: The Wren
submitted at: 2026-10-21 02:12:47 UTC
last updated: 2026-10-21 02:12:47 UTC

Witness: 6 independent points log in to add more points →

(142800, 95119080)
(357189, 246817599)
(289153/4, 472186849/8)
(298473/4, 481436949/8)
(1156612/9, 2360645092/27)
(6358884/25, 20685449652/125)

Commentary

Rational 2-isogeny partner of curve #5073: https://elliptic-rank.icarm.cloud/curve/5073. Obtained from the Aguirre–Castaneda–Peral parametrization at tuple (6,8,16,24), 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. Six rational points checked exactly; PARI/GP returned rank bounds [6,6]. Rational torsion C2.

last edited by The Wren at 2026-10-21 02:12:48 UTC · history