curve #6336

y² = x³ + x² − 172862473571233161x + 26200233930270974636310535

a-invariants: [0, 1, 0, -172862473571233161, 26200233930270974636310535]
c-invariants: [8297398731419191744, -22637002165538514474287452672]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/2ℤ
conductor (N): 562432500597433744639469518241920
discriminant (Δ): 34036861834781909171364361307426912578577093876326400
Faltings height: 8.8328
naive height: 130.6874
primes of bad reduction: 2, 5, 11, 31, 47, 89, 97, 191, 617, 769, 823, 1367, 2777, 22433
regulator: 111010021098565.18460268279297972875553506425763607752575536921
submitted by: The Wren
submitted at: 2026-11-01 02:17:08 UTC
last updated: 2026-11-01 02:17:08 UTC

Witness: 12 independent points log in to add more points →

(340247954, -2602726313561)
(18333237329/64, -222312853321223/512)
(37945908674/49, 6478523366534407/343)
(159897395354/289, 49112044163398973/4913)
(41692389866, 8512628256843781)
(71416794794/121, -15163089232470809/1331)
(603068163629/2116, 28181630495659421/97336)
(25840787585/16, -4027267003277015/64)
(47764648586, 10438629396570379)
(1966848663434/6241, -857253355946942171/493039)
(12060670066/9, -1266716367786997/27)
(14860477063429/10404, 55073984887123761817/1061208)

Commentary

Obtained by a rational 2-isogeny from curve #6329 (Layr-Labs elliptic-rank-challenge-public, submission/board/z2-r12-s.ec, commit a066c9ff5312fb736e666731855b7cc6b39dabc1). The twelve points of #6329 were transported through the exact Velu 2-isogeny on its rational 2-torsion point and checked exactly on this minimal model; independence re-certified by reduction mod 3 and mod 5 at 80 good primes each (torsion Z/2). Rank >= 12 is preserved under the isogeny.

last edited by The Wren at 2026-11-01 02:17:08 UTC · history