curve #6442

y² = x³ − 6228687308306417892x + 5406090867797520081415191300

a-invariants: [0, 0, 0, -6228687308306417892, 5406090867797520081415191300]
c-invariants: [298976990798708058816, -4670862509777057350342725283200]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 68703648682458326225449551581699878684569624 ★ record for rank ≥ 17
discriminant (Δ): 2840145690292606695657045392520724522361316133781151282432
Faltings height: 9.7588
naive height: 141.4407
primes of bad reduction: 2, 3, 7, 11, 43, 47, 801833, 7959004901, 137261749107944340721
regulator: 11560529572165.5972492292136214106607116458769008061633749607987710443595
submitted by: Michael Rubinstein
submitted at: 2026-11-03 14:18:05 UTC
last updated: 2026-11-03 14:18:05 UTC

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

(45023090784, -9538884622518174)
(24928965513, -3916927339201701)
(-89797833912/49, 35423952691552326/343)
(-2983328768/9, 2328013714282102/27)
(869981806, 25410783578158)
(2730309729/4, -306990601736049/8)
(17428023573822/9409, 13654991291789063502/912673)
(6120745268400/2809, 6950459512983439530/148877)
(38063147017/16, 4073818242752027/64)
(18110030464/9, -862383834765046/27)
(50815820454/25, -4226912666988858/125)
(5262936064488/2401, -5623670778651983574/117649)
(5825994898794/2809, -5589059431838376186/148877)
(3807786192, -192090006957618)
(1142782207974246/573049, -13125692423465636335398/433798093)
(-1006074412334/361, 229476981278974022/6859)
(67215666, -70623857008782)

Commentary

Mestre ten-root construction, roots 0,2,14,18,33,44,97,106,120,136: 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 14:18:05 UTC · history