curve #6503

y² = x³ − 356484526991117814367x + 2586738858832553806989380365090

a-invariants: [0, 0, 0, -356484526991117814367, 2586738858832553806989380365090]
c-invariants: [17111257295573655089616, -2234942374031326489238824635437760]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 273533793796876732889999722911159668751149628173587165378376
discriminant (Δ): 8753081401500055452479991133157109400036788101554789292108032
Faltings height: 10.5872
naive height: 153.5821
primes of bad reduction: 2, 419, 1013, 134764445027, 594452802907, 381063290769779, 2638807524349021
regulator: 856708172309491669.5088070501031045033668742197602488434983928708639439596085
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:54:35 UTC
last updated: 2026-11-04 04:54:35 UTC

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

(402733330137, -255303746030590358)
(170345774049/4, -64319365597065209/8)
(123654481309/49, 447637919646192794/343)
(960059166849/169, 1896232896078262286/2197)
(9934947665829/1225, 20518964160928021858/42875)
(50752476961830/5329, -91633285874834830400/389017)
(560995291753/49, 26731079415399206/343)
(3659111421521/256, 2627589300752223193/4096)
(470415904454646/22801, 6897760971606782585788/3442951)
(51752072150781/3481, -155452506796272758122/205379)
(-11981068159, -2266709053565042)
(202759121641/25, -59817822928847686/125)
(-329199492123/49, -741902518612874578/343)
(26130191071, -3333628928233712)
(6753348066949/169, -15866028281402067586/2197)
(-1061971472931/49, -124961449068042586/343)
(-52304369553467/2401, 14177816731506002546/117649)
(-49308454015288613/2362369, -3509394808249363728747088/3630961153)
(19139155423759409/894916, -1844058008543221317839599/846590536)

Commentary

Mestre ten-root construction, roots 0,4,21,26,35,73,84,96,151,170: 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 >= 19: the 19 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 04:54:35 UTC · history