curve #6173

y² + xy = x³ − 503897620966323620x + 151217417043830260627562256

a-invariants: [1, 0, 0, -503897620966323620, 151217417043830260627562256]
c-invariants: [24187085806383533761, -130651848362149973891789089825]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 2206369951742974882400755286879288413262834238
discriminant (Δ): -1689869548427228196035762708318731556714076077720629248
Faltings height: 9.1408
naive height: 134.0847
primes of bad reduction: 2, 3, 19, 109089012578454890341, 177415872838624908605387
regulator: 49108554686216799909.3828171496952221845606290244191053930535068278937549
submitted by: Michael Rubinstein
submitted at: 2026-10-30 14:39:56 UTC
last updated: 2026-10-30 14:39:56 UTC

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

(2729912911/4, 89511730498685/8)
(4115379736, 260338105334428)
(-987992467973296856/1298232961, 453382296848172943374313060/46776631817791)
(-6517336747208/7921, 2091669941071973756/704969)
(38986164244/9, 7600920370335812/27)
(134280730696/289, 20491190196003740/4913)
(3125322374536/4489, 3531334755784887988/300763)
(5912591814184/16129, 8131429671468659908/2048383)
(6655357818136/14641, 7105793877283411708/1771561)
(78705008295652/80089, 557513846236397041828/22665187)
(149067441950500/426409, 1174205178846771668732/278445077)
(171623317705480/22201, 2239210210041517946924/3307949)
(245094630455604/112225, 3658026373062048007872/37595375)
(1237548612228514/2486929, 19088446110347041731974/3921887033)
(3175527649113052/3038049, 146603015432508595496488/5295319407)
(10631995715997316/990025, 1093945966761634175085104/985074875)
(459938989482922888/103286569, 308204464462005487822534388/1049701400747)

Commentary

Mestre ten-root construction, roots 0,5,19,29,40,50,101,105,118,131: 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 14:39:56 UTC · history