curve #6133

y² + xy + y = x³ − x² − 83944040996062112x + 9499994324931186677586755

a-invariants: [1, -1, 1, -83944040996062112, 9499994324931186677586755]
c-invariants: [4029313967810981361, -8207995078608632434285540425]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 66990644961370594294331709277299659581362
discriminant (Δ): -1130657327131201185816591008387567774008199833179648
Faltings height: 8.5942
naive height: 128.5498
primes of bad reduction: 2, 3, 7, 113, 28031, 39359, 4264641446277104227915783831
regulator: 66192228708206164744.01302663565167163332924261104194115541626854222656
submitted by: Michael Rubinstein
submitted at: 2026-10-29 12:56:18 UTC
last updated: 2026-10-29 12:56:18 UTC

Witness: 16 independent points log in to add more points →

(32753253111, 5927390347072807)
(-937402001232768947/3887148409, 960888584814765588277706875/242352041855923)
(-15056216822864753/44930209, 760568644632612905919/301167190927)
(-2607355206899/15625, 8482470823398239349/1953125)
(-842074938723/3481, 812520832013488475/205379)
(-5031536411/49, 1415782073681151/343)
(142644918345/529, 30949101087330815/12167)
(215440281789/1681, 63413527868740745/68921)
(8688404949349/55696, 5894484992382170075/13144256)
(118353449053797/11449, 1287064961429413248611/1225043)
(251594255169283/1515361, 696676298090918802429/1865409391)
(626228432117797/12737761, 106535702907226094394105/45461069009)
(1540707762559143/23843689, 242704239418814975499989/116428733387)
(7820426705091153/17447329, 573488401815656517972421/72877493233)
(2393739263501443317/29739347401, 9266447711246494693432832075/5128580198649851)
(64533457481690024085/493974225889, 303261427280035062248193941825/347181387104243537)

Commentary

Mestre ten-root construction, roots 0,8,13,24,66,72,95,107,116,139: with p(x) = prod (x - a_i), write p = q^2 - r with q a monic quintic and deg r <= 4; the genus-one curve y^2 = r(x) carries the ten points (a_i, q(a_i)), and the curve submitted is the minimal Weierstrass model of its Jacobian. Found by enumerating integer ten-root sets 0 = a_1 < ... < a_10 and scoring each by S = sum of -a_p log(p)/p over the good primes 5 <= p <= 199 (with early abort on low partial sums); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 16, proved: 16 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-29 12:56:19 UTC · history