curve #6519

y² + y = x³ − x² − 6674202848731630x + 162655156454237295317142

a-invariants: [0, -1, 1, -6674202848731630, 162655156454237295317142]
c-invariants: [320361736739118256, -140534053254290602719301400]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 133299129597562732119902707030702830923252701821
discriminant (Δ): 7598050387061075730834454300750061362625404003797
Faltings height: 8.0915
naive height: 120.9247
primes of bad reduction: 3, 19, 20051, 50363, 16062394022628251, 144176524948432809631
regulator: 289228768909796.132825990042882451536063144491613105588244656028195671145
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:56:29 UTC
last updated: 2026-11-04 04:56:29 UTC

Witness: 17 independent points log in to add more points →

(67130623210/169, -17044473347809454/2197)
(2558519358640, -4092446972518509107)
(423545399869/900, -271678031936179253/27000)
(10767162129549/625, -35330273748906431157/15625)
(-130519816780/22201, 1485595594224188369/3307949)
(579992286377/23104, -367130259446973685/3511808)
(2701153030221/24025, 3399707111296793281/3723875)
(102696684549/676, 28637889404042745/17576)
(34238904821041/30625, 199821682571917682536/5359375)
(1332398610862642/4313929, -47042045043313216906430/8960030533)
(42466523433/121, -8527767502633371/1331)
(15648407479053676/53246209, -1886879949710252035357883/388537587073)
(786005229124/729, -694890240112939906/19683)
(16507695908094/175561, -44528138582075762570/73560059)
(-1832236067/36, -131477276264005/216)
(4673003443821454/43678881, -236845320885056808024659/288673724529)
(3017550840270/16129, -4787471514194022402/2048383)

Commentary

Mestre ten-root construction, roots 0,13,60,75,149,152,155,156,175,191: 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 >= 17: the 17 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 04:56:29 UTC · history