curve #6487

y² + xy = x³ − 842090846325723626475635x + 387220554562801613280889701177360225

a-invariants: [1, 0, 0, -842090846325723626475635, 387220554562801613280889701177360225]
c-invariants: [40420360623634734070830481, -334558559142321224415624153918345480121]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 4696433430798031027581017502847138013984755935928291846619310
discriminant (Δ): -26556955935454327797485778256675731827073300081136450588779507916032000000
Faltings height: 12.7975
naive height: 177.4118
primes of bad reduction: 2, 3, 5, 101, 51421, 3488029, 88258214593, 898391459550157, 108989380645966881553
regulator: 1121320570323967637.42088114845475370533542412555879763706621262352410808886213
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:52:42 UTC
last updated: 2026-11-04 04:52:42 UTC

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

(125778359089204/25, -1389133340141236081177/125)
(11251687731070, -37621559406100118135)
(-35795849897870/169, 1638321732037423341505/2197)
(827440287600250/2809, 60424152446643780015905/148877)
(225433650519670/361, 2221298789429895706435/6859)
(110122661109400/169, 745614381371793555955/2197)
(4983521860634530/6889, 226282112782160431516655/571787)
(6473381944682170/7921, 348836853889191393137885/704969)
(18475250636787970/14161, 2070261395838547833820235/1685159)
(744484747950, -415852567137910215)
(294446353829314966/358801, -107208547897369695133101817/214921799)
(96297457021750420/120409, -19831328484254630151844405/41781923)
(1224491690157280/1369, -29949488739705918925805/50653)
(366463083689350/289, -5725962645001521408895/4913)
(675713504725630/1369, -15349435541547311067755/50653)
(3682180717404828070/4330561, -4818942612358659806995433335/9011897441)
(13696509363166/25, -37565223806477341579/125)
(-534584245095770/8649, -532963453307239473788795/804357)
(2262327724456650/3481, -69479857179857547755565/205379)
(-21410873090, -636585128544467975)

Commentary

Mestre ten-root construction, roots 0,10,39,53,76,78,83,89,119,165: 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 04:52:42 UTC · history