curve #6513

y² = x³ − 1668750035846556247x + 944476357300590618511075930

a-invariants: [0, 0, 0, -1668750035846556247, 944476357300590618511075930]
c-invariants: [80100001720634699856, -816027572707710294393569603520]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 16457814113570642745548909115192863789336093133235112
discriminant (Δ): -87950558622921514832213370311590664090212081704008438528
Faltings height: 9.4611
naive height: 137.7485
primes of bad reduction: 2, 167, 1151, 10702626532493641786073103000250279494878245117
regulator: 8326811878457588.6124603595615606019541663616734195792091092565318091516167
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:55:47 UTC
last updated: 2026-11-04 04:55:47 UTC

Witness: 18 independent points log in to add more points →

(-368531598, -38851141001962)
(135232794709, -49728264461906806)
(-9094395893/81, -24510259289203748/729)
(16593077562533/3721, -65068028665245088474/226981)
(14945740518963/5929, -51561764502528367396/456533)
(50366415217/64, -5576072440430473/512)
(174018280097/289, 61750571479591198/4913)
(13753557056165/19321, 29085816658297382330/2685619)
(2241877944549/2209, 1786865984395201574/103823)
(-1472356933, 14479131232012)
(-997308503, 40209299116762)
(2694725677/9, -586385906237342/27)
(36413534147/121, -28841258822331572/1331)
(-660211813, -41933661393212)
(-29462477714187/22201, 94833183650937644746/3307949)
(-7515303781425485/5517801, 340639138875553911489340/12961314549)
(-2148469115282/1521, 1304029283660326124/59319)
(6483259333/9, -291028806684026/27)

Commentary

Mestre ten-root construction, roots 0,12,27,61,77,96,136,139,188,200: 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 >= 18: the 18 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 04:55:47 UTC · history