curve #6493

y² + xy = x³ − x² − 268236005824488436622434x + 53696023198846354788945604851827344

a-invariants: [1, -1, 0, -268236005824488436622434, 53696023198846354788945604851827344]
c-invariants: [12875328279575444957876841, -46393364043745311560390913089668379445]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 15141241539563545332571561817468094769657291789078778794197996638938
discriminant (Δ): -10386891696140592098144091406783113011984902167308042252819825694311468
Faltings height: 12.2835
naive height: 173.4604
primes of bad reduction: 2, 7, 97829917669660331537, 11055076796424360976352235184081525568192325691
regulator: 43160327638106075202.2980752951799132595670937171201881568299089958658725751524
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:53:25 UTC
last updated: 2026-11-04 04:53:25 UTC

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

(3354307890078, -6074089867875408098)
(129586837313350, -1475155802140200997762)
(-438095411359694/1521, -19409248468372185086054/59319)
(50400792824241/256, 378018853391388608239/4096)
(844584534257246/4225, 24660326000631354819486/274625)
(742101740308864/3249, -12206930720045775434188/185193)
(242799558515961/961, -1349876203969024866626/29791)
(1767583330903774/5625, 8805356800071478014332/421875)
(7192427499933378/22801, 74704263117277885479802/3442951)
(1420470284519/4, -454246799007164085/8)
(22838018146410413/38416, -2432146598318551059827645/7529536)
(45656578853, -203824597021926098)
(-25374348664651262478/44448889, -42722999741305872468806617150/296340742963)
(-8195469404745557/16641, -552951981342905655733972/2146689)
(1047472748285101/400, -33283153030022429918511/8000)
(884800433571092933/737881, -764877012586513088440317630/633839779)
(3844135683097, -7471869888989119227)
(8605433269055, -25199363510831912847)
(8860738291627, -26331682391635300567)
(109670092635502007/58564, -35056414517872720818061345/14172488)

Commentary

Mestre ten-root construction, roots 0,1,39,64,65,114,124,150,151,168: 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 04:53:26 UTC · history