curve #6507

y² + xy + y = x³ − x² − 1400455092494322335477x + 20196101244440347233074584760845

a-invariants: [1, -1, 1, -1400455092494322335477, 20196101244440347233074584760845]
c-invariants: [67221844439727472102881, -17449431474893961709397667608907345]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 50386897596845092487449703156078065833519331942554
discriminant (Δ): -417926274943831303860498758350837905008872589937060969370288128
Faltings height: 10.9205
naive height: 157.6892
primes of bad reduction: 2, 3, 11, 13, 41, 476122015347169, 77137152711882516579848849023
regulator: 9359673150175581.5251386652917547962895854892970966980134140822639301694753
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:55:04 UTC
last updated: 2026-11-04 04:55:04 UTC

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

(151089110623, -57075925241481328)
(33749720650591, -196067243990509759984)
(-1614436001177/121, -8041959225514855376/1331)
(26237254577743/1521, 64195761425705433328/59319)
(34422059675527/1849, 60366791367160585904/79507)
(1071213886219/49, 57609912776124428/343)
(334549119776911/14161, 917324061339761594224/1685159)
(24057082300111/961, 27069302511279159776/29791)
(1422590310943/49, 686717324276509856/343)
(28343293535, -1808869545689200)
(2795969649596695/94249, -63115493834990247059920/28934443)
(1244932703, -4295877525837296)
(1588048159279/49, -1020370506585638032/343)
(-1523953496064353/173889, -408881406329107132340464/72511713)
(28572674647, -1872985304623120)
(213705866383207/10201, -234086885442042443408/1030301)
(863305992109591/3721, -25054370547651156781744/226981)
(2059141531183/49, -2045268646151251984/343)

Commentary

Mestre ten-root construction, roots 0,1,55,78,86,112,119,124,140,165: 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 >= 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 04:55:04 UTC · history