curve #6448

y² + xy = x³ − 226606766226198211050x + 1286638541856085297831381102500

a-invariants: [1, 0, 0, -226606766226198211050, 1286638541856085297831381102500]
c-invariants: [10877124778857514130401, -1111655700179973384494599543755601]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 594031745999888841588209639676759504494404410
discriminant (Δ): 29580027970070802426245383657148505349573710750202296000000000
Faltings height: 10.5815
naive height: 152.2228
primes of bad reduction: 2, 3, 5, 53, 179, 178469, 9239887, 24203837090149, 52293273415723
regulator: 17365458542869.3058067879202587419238943525085984735958563148616489156066
submitted by: Michael Rubinstein
submitted at: 2026-11-03 14:23:48 UTC
last updated: 2026-11-03 14:23:48 UTC

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

(1171022451900/121, -3865049063173350/1331)
(30627178819900/9, -169494795365705871350/27)
(95025112058350/9801, -30275567400150136450/970299)
(2491822620084/25, -3890939782364058162/125)
(41327457676950/5929, -97668193049249706750/456533)
(4078023623700/1681, -59734313544543064650/68921)
(7281948247860/361, 15219047399183976450/6859)
(159745537158900/2809, 1954317901589064759150/148877)
(127037931661165/1296, 1415860515231431723675/46656)
(9776035830, 75008373370320)
(-1612572900, -1283691193921050)
(135209184004/9, -30463373553959086/27)
(5319042816900/529, 1911779068498996650/12167)
(9862291860, 105020439259590)
(-5128857900, 1521172146538950)
(232137344262460/23409, 433589230372806686350/3581577)
(-436921878983025/217156, 133271266891275462009825/101194696)

Commentary

Mestre ten-root construction, roots 0,22,33,50,77,82,95,106,120,143: 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, proved: the 17 listed points come from the point search and are independent, and the 2-descent upper bound (ellrank) is 17. Torsion trivial.

last edited by Michael Rubinstein at 2026-11-03 14:23:48 UTC · history