curve #6489

y² + xy = x³ − x² − 5146779502813002917578x + 147174015514176512090179370677000

a-invariants: [1, -1, 0, -5146779502813002917578, 147174015514176512090179370677000]
c-invariants: [247045416135024140043753, -127158349403136802073307367634731125]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 31737968087921775307522975461902370443739390331195531141429586
discriminant (Δ): -631775992758170859271552349544628586053076303932778242901297338916
Faltings height: 11.3927
naive height: 161.6615
primes of bad reduction: 2, 37, 269, 2081, 417581948294786754772603031, 1834768700946805932206043071
regulator: 20254320983398498603.7064329186707485602436402149914632427818981608835839898442
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:52:57 UTC
last updated: 2026-11-04 04:52:57 UTC

Witness: 20 independent points log in to add more points →

(9289622161606/625, 134373998654294956554/15625)
(695051778182, -576496060284340614)
(2232136032572/169, 19833912196471111692/2197)
(1003478511477/16, -539573957469023897/64)
(146077480847950/2401, -905558115117645069270/117649)
(-12598166070139/400, -133394145238281344619/8000)
(137159916939394/2025, 952732291645556476082/91125)
(100228150955951/1225, 709437738126138665496/42875)
(62913357737246/625, 398073438945696488344/15625)
(43095547097410700/1142761, 3165030610212200139034080/1221611509)
(7359188560950/361, -48851207876006514370/6859)
(-21423360006, 15735411970720588)
(-99771831139956/1225, 218978356490301610208/42875)
(5490544241263/169, 8294828093497417224/2197)
(-2719981932293/36, 2210482933970425727/216)
(558025827370437676/16152361, 211340326478193103124861484/64916338859)
(7185845623010063/208849, 313969641521172930966288/95443993)
(8432497473311521/326041, 1042571697844854657724671/186169411)
(11774270481040591/514089, 2369116584573373495543784/368601813)
(137728761388924066/4748041, 48842040512737714558111662/10345981339)

Commentary

Mestre ten-root construction, roots 0,25,26,48,49,80,135,140,150,175: 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 >= 20: the 20 listed points come from the point search and are independent (height matrix nonsingular at 57 and 115 digits). Torsion trivial.

last edited by Michael Rubinstein at 2026-11-04 04:52:57 UTC · history