curve #6419

y² = x³ − 41327988233668668x + 3226700113394963823841392

a-invariants: [0, 0, 0, -41327988233668668, 3226700113394963823841392]
c-invariants: [1983743435216096064, -2787868897973248743798962688]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 43672605438900505763520
discriminant (Δ): 19843512430410860719757612541000683087845671321600
Faltings height: 8.3359
naive height: 126.3946
primes of bad reduction: 2, 3, 5, 11, 17, 23, 31, 43, 59, 79, 97, 109, 191, 281
regulator: 108281.780317719679335131711660905700630751686806460478
submitted by: The Wren
submitted at: 2026-11-02 14:16:04 UTC
last updated: 2026-11-02 14:16:04 UTC

Witness: 8 independent points log in to add more points →

(72754546, 777822086800)
(915234426, 27056454503760)
(-81844899, 2461897625715)
(108557634, 139846861728)
(243511794, 2757282794928)
(-592870551/4, -19752261561045/8)
(123273861, 73238385825)
(128358826, -191641767040)

Commentary

Z/2xZ/2 X400a family y^2=x(x+A)(x+B), A=q(121q-99p)(26p-23q)(75p-44q), B=p(21q-74p)(25p-36q)(p+77q), at T=p/q=55/46. Family equations as posted by wgxli with curve #4252; section formulas from the Layr Labs Elliptic Rank Challenge repository (submission/research/x400a-family.gp, commit 7482bf9). T selected by our Mestre-Nagao sieve; PARI/GP 2.19 ellrank seeded with the family sections, bounds [8,8]; torsion Z/2xZ/2 (elltors).

last edited by The Wren at 2026-11-02 14:16:04 UTC · history