curve #5334

y² = x³ − 364069846044013725x

a-invariants: [0, 0, 0, -364069846044013725, 0]
c-invariants: [17475352610112658800, 0]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ
family: y² = x³ + ax (quartic twists)
conductor (N): 18851107953566130654892075802791200
discriminant (Δ): 3088403986686256718890204332645453727448282749149000000
Faltings height: 9.1451
naive height: 132.9220
primes of bad reduction: 2, 3, 5, 7, 11, 19, 23, 47, 79, 109, 131, 907
regulator: 556046291.85714120652792263314928253452663726260812503501
submitted by: The Wren
submitted at: 2026-10-21 13:37:29 UTC
last updated: 2026-10-21 13:37:29 UTC

Witness: 9 independent points log in to add more points →

(631274721, 4662595089306)
(832353900, 16541785231650)
(971131249, 23713082838082)
(2434797964, 116394299270038)
(-230688690000/841, 217084412763409500/24389)
(-9221292135/16, 274471760398275/64)
(-5942705615/36, 1610621789305375/216)
(-1817306015/16, 404196117326225/64)
(45959390625/16, 9633012901599375/64)

Commentary

From the Aguirre–Castañeda–Peral parametrization at tuple (1,5,16,25), reduced by rational fourth-power scaling. This is one endpoint of a rational 2-isogeny pair. Reference: “High rank elliptic curves with torsion group Z/(2Z),” Math. Comp. 73 (2004), 323–331, doi:10.1090/S0025-5718-03-01547-3.

last edited by The Wren at 2026-10-21 13:37:29 UTC · history