curve #6467

y² + xy + y = x³ − x² − 91959660554425588454627x + 10361689527741074403357566076135715

a-invariants: [1, -1, 1, -91959660554425588454627, 10361689527741074403357566076135715]
c-invariants: [4414063706612428245822081, -8952499751948424997821181162675058625]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 8845203848788098026820445114527786174959429151384172062
discriminant (Δ): 3388994127650467866726326437178360513046946849552588945269282114895872
Faltings height: 12.1085
naive height: 170.2405
primes of bad reduction: 2, 3, 17, 31, 47, 1303, 5113, 208433, 303978118223363606172009310463062199
regulator: 1032205263562509811.096850127138854191656998309341196291382507158600602085905
submitted by: Michael Rubinstein
submitted at: 2026-11-04 02:10:08 UTC
last updated: 2026-11-04 02:10:08 UTC

Witness: 19 independent points log in to add more points →

(4925556853351/36, -3988495411852541683/216)
(174477101258425, -2304660810159862633816)
(24151699327211/169, -25850188952848275430/2197)
(204807933681517/3721, 16782590234701652321030/226981)
(-64391137493/4, 870415680283555537/8)
(-197001266073959/576, -587483439992168604871/13824)
(-1758309099083255/5329, -26903682509109249679762/389017)
(4790775447595/4, 10176922165303524865/8)
(461656532072467/361, 9661580058506285140838/6859)
(-277956617765, 120198268891533170)
(-275873735825, 121388812577141390)
(-111259342133, 138621198988317122)
(203824261987, 9263911185350762)
(232859003419, 39679120199138930)
(254091013531, 58311552514445042)
(436001149741, 230541933202412996)
(500261444123, 299255497645657714)
(-175370207942989/841, 3489447290843599867834/24389)
(51390035839019/625, 905210665760157366722/15625)

Commentary

Mestre ten-root construction, roots 0,18,39,61,62,72,73,94,95,116: 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 19, proved: 19 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-11-04 02:10:08 UTC · history