curve #6488

y² + xy = x³ − 311310556984431516061386470x + 2099176015151900501826081277450638173412

a-invariants: [1, 0, 0, -311310556984431516061386470, 2099176015151900501826081277450638173412]
c-invariants: [14942906735252712770946550561, -1813688077091264447937837102786507801653809]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 4717721772249207926330637464217482024447812014820011301605990
discriminant (Δ): 27282467042020061927279922914105390091247496570795617309974922879330114560000000
Faltings height: 14.0689
naive height: 194.6221
primes of bad reduction: 2, 3, 5, 7, 239, 1619, 37588753211, 4826273305783, 320035715605441459690577420243
regulator: 636538277768158637.064614801752982495143144525870534468678414934350760107505562
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:52:50 UTC
last updated: 2026-11-04 04:52:50 UTC

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

(82235123943514, -729810322387788534632)
(47957538947884, -312199089653860687862)
(1600361668734, 40063265553494951028)
(543432530205148/169, 73898330346018982513666/2197)
(677088943423324/81, 6560429865743784653002/729)
(-955307785073732/1849, -3779619589051881678351314/79507)
(260221418384939164/22801, 19713563875246807984363318/3442951)
(2124039388196166/169, 28800321086186901689196/2197)
(11298922759908024/841, 450866496647263888006902/24389)
(187405249071957004/17161, -3061639905297371852827762/2248091)
(10985616179884, -2241259875581044022)
(3222762733249036/11449, -54943799476017006740599426/1225043)
(53786750391149116/1369, -11384643768217765102084046/50653)
(54986427250945324/5041, 404254278547560523224998/357911)
(-99216020263316852/5329, -14769863649849656343422150/389017)
(8183186881011196/529, -381887981392145401158634/12167)
(-108402184934276/25, -7253792038192455613342/125)
(16119331689686839/676, -1594879828257209476719907/17576)
(21278567494264, -71479997480824267382)
(11994959793904, -9531110134813527062)

Commentary

Mestre ten-root construction, roots 0,4,12,13,27,43,151,156,174,184: 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 04:52:50 UTC · history