curve #6449

y² + xy + y = x³ − x² − 462229216700775066822x + 3292992848523606943193733248085

a-invariants: [1, -1, 1, -462229216700775066822, 3292992848523606943193733248085]
c-invariants: [22187002401637203207441, -2845145821024554888112018111912185]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 831896611260754035997450667257493457847065982
discriminant (Δ): 1635988019109418827711762446319038294652539423454037724393439232
Faltings height: 10.8539
naive height: 154.3614
primes of bad reduction: 2, 11, 17, 19, 23, 43, 443063, 683290872278032605040564338731
regulator: 41586334606869.4633418060307616199720175399776032242227561167120888372857
submitted by: Michael Rubinstein
submitted at: 2026-11-03 14:23:55 UTC
last updated: 2026-11-03 14:23:55 UTC

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

(118977449001, -40404258369912709)
(690930774761913, -18161504121493260105333)
(-43322301054071/2025, -167779641255275087857/91125)
(-3854400103703/9409, 1703126373223139424987/912673)
(26919878350513/3249, 33109652196883593355/185193)
(210295576129/25, 444897157891407/125)
(67742426212289/4225, 16674580881594796883/274625)
(2619433819117/81, 3432069573793511275/729)
(1059927306588977/24649, 30698762449535221500967/3869893)
(84595053396952113/2111209, -21496339141827318127653225/3067586677)
(268483609309/9, -108174511270133843/27)
(33047483789, -4910193013851005)
(44063306005, -8275156832229581)
(53985770319865/1681, -320107550068739115741/68921)
(-20282379035, -2079518504660345)
(-604252836101/49, -915092896253694537/343)
(2311573806707113/47089, -101618278280495585909469/10218313)

Commentary

Mestre ten-root construction, roots 0,1,90,97,114,115,130,144,157,172: 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); for the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, rational points were found by a point search on the quartic y^2 = r(x), and the curves were screened by conductor and by a GRH explicit-formula upper bound on the rank. Rank 17, proved: the 17 listed points come from the point search and are independent, and the 2-descent upper bound (ellrank) is 17. Torsion trivial.

last edited by Michael Rubinstein at 2026-11-03 14:23:55 UTC · history