curve #6385

y² + xy + y = x³ + x² − 14218712251970551130824526x + 12389472280990399517322401765150815898

a-invariants: [1, 1, 1, -14218712251970551130824526, 12389472280990399517322401765150815898]
c-invariants: [682498188094586454279577249, -10704504050780823919377264523497401765393]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 16334391992898237107214324401202491115
discriminant (Δ): 117664251543947196560280847013585040596442512818419075008341516024809388962225
Faltings height: 13.4885
naive height: 185.3634
primes of bad reduction: 3, 5, 7, 11, 19, 47, 61, 109, 127, 163, 167, 367, 443, 463, 619, 1301, 1567, 7253
regulator: 3966245035.41239331220504948779332146255565929032253401976279
submitted by: The Wren
submitted at: 2026-11-02 01:03:52 UTC
last updated: 2026-11-02 01:03:52 UTC

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

(-290381469522416/81, 3031220638090564345873/729)
(1171616914103/4, 22978082908797488287/8)
(285021352442347863/51076, 119301331401147978120559483/11543176)
(-190414672197591/64, 2726452632817614364703/512)
(112876407191931, 1198572068641703603194)
(-3487642781898, -4422303958848836462)
(1907428224122664/529, 34412663918867640144218/12167)
(-11062540169801/4, -44224597866597263573/8)
(-1179768336206394/289, 7618028212377332440946/4913)
(11186130725152, 35398472810626016438)
(3298998366816, -1177392985786217546)

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=-2672/61. 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 [11,11]; torsion Z/2xZ/2 (elltors).

last edited by The Wren at 2026-11-02 01:03:52 UTC · history