curve #6178

y² + xy + y = x³ − x² − 685990640466912207377x + 6868986816410030691695751537025

a-invariants: [1, -1, 1, -685990640466912207377, 6868986816410030691695751537025]
c-invariants: [32927550742411785954081, -5934804609230092539284276291196465]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 4611183920987854374283972944656606152532275998
discriminant (Δ): 277153811365506991014369371472342991972265109964043691252252672
Faltings height: 10.8104
naive height: 155.5458
primes of bad reduction: 2, 3, 11, 13, 152567, 9767749697, 1202122631955592866183495823
regulator: 40502905044105895411.5027022790282188857048388954134323457620087222879967
submitted by: Michael Rubinstein
submitted at: 2026-10-30 14:40:33 UTC
last updated: 2026-10-30 14:40:33 UTC

Witness: 17 independent points log in to add more points →

(-3565807912929857/120409, 46008312357622292196700/41781923)
(-854558734918073/28561, 3764667787884528903188/4826809)
(-67342583129405/2401, 236620266419772723956/117649)
(-24189745337365/253009, 335130383348272138560140/127263527)
(-573570712001/25, 405646334748315172/125)
(535554875569/25, 176971048928188312/125)
(1143081337967/361, 14915109440400131708/6859)
(138809286401051/3481, 1345536527979218981084/205379)
(142817909590639/361, 1703115168641738154620/6859)
(172327598946439/10609, 119540516382484643564/1092727)
(516613501506143/32041, 8263025191609916/5735339)
(886236605174311/54289, 1825457758622833914580/12649337)
(1801767479174053/54289, 57492655939796088460450/12649337)
(2915178059021359/218089, 30166376971577986110076/101847563)
(3949182514516879/126025, 179759080740677348982332/44738875)
(85194132867523055/537289, 24546592624676536063106020/393832837)
(10254696380517660847/907877161, 20500413118847034521599407100/27355246738091)

Commentary

Mestre ten-root construction, roots 0,1,22,28,54,82,96,104,134,137: 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 17, proved: 17 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-30 14:40:33 UTC · history