curve #6396

y² = x³ − 12186792637310344084024260708x + 514652203854810321145615636415113965835968

a-invariants: [0, 0, 0, -12186792637310344084024260708, 514652203854810321145615636415113965835968]
c-invariants: [584966046590896516033164513984, -444659504130556117469811909862658466482276352]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 79295595036121358250617842517352625894080
discriminant (Δ): 1414753063229824692904474740434030911760271367703886547083533970799476536860285440000
Faltings height: 14.9796
naive height: 205.6241
primes of bad reduction: 2, 3, 5, 11, 17, 23, 37, 43, 61, 89, 151, 163, 173, 179, 263, 443, 479, 859, 1663, 12197
regulator: 4874422901.46907885639576714447371141468216109933214071214869
submitted by: The Wren
submitted at: 2026-11-02 01:17:08 UTC
last updated: 2026-11-02 01:17:08 UTC

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

(52062025402461, 145927538372663116569)
(2252746958713129/16, 80698502119676560066475/64)
(58190764672194, 50368218789774146400)
(44641574795712589/625, 1448337472624310305557313/15625)
(104263297876794, 614366462584492479000)
(58937640177594, 33464442599163714600)
(36520155928806, 343942384957838742456)
(-126127430690622, 212813623559043216864)
(166613843530857/4, 2252810768239220196843/8)
(-27772451761956, -911969391140150440500)
(176717613287577372/1369, 52942979478598899026977332/50653)

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=1916/575. 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 01:17:08 UTC · history