curve #4304

y² = x³ + x² − 1538229063973728558x − 685301729300535025882283112

a-invariants: [0, 1, 0, -1538229063973728558, -685301729300535025882283112]
c-invariants: [73834995070738970800, 592100693672652291937858784000]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ
conductor (N): 413720850995012265670517042400
discriminant (Δ): 30055617704316686393875235649686778792446040712977000000
Faltings height: 9.3908
naive height: 137.2451
primes of bad reduction: 2, 3, 5, 7, 13, 23, 31, 37, 47, 61, 101, 109, 409, 1217, 3137
regulator: 1280247923219.92339970708732513621508639220571823889547430183
submitted by: The Wren
submitted at: 2026-10-16 19:15:52 UTC
last updated: 2026-10-16 19:15:52 UTC

Witness: 11 independent points log in to add more points →

(-103772082762/121, 2352550674873000/1331)
(-648030522, 6275461071600)
(-7755254458106/12321, 7920251161350631424/1367631)
(-5955060966354/10609, 1234916259780048468/1092727)
(-790320306, -6062423917284)
(-3176769963/4, -47610359497575/8)
(467112845637/4, -319233255997905375/8)
(2211934278, -82063780780800)
(5798467581/4, 91586772173043/8)
(6318711573/4, 230018016124089/8)
(3934666017318/1681, -6368435600522797200/68921)

Commentary

Obtained by a rational 2-isogeny from my curve#4254, found in a point-constrained cover search using the X1008 family associated with curve#4118. This isogenous curve has the same rank and conductor. The eleven supplied points were transported from the source curve.

last edited by The Wren at 2026-10-16 19:15:52 UTC · history