curve #6445

y² = x³ − 2315798183384130627x + 1415442519060208520102098146

a-invariants: [0, 0, 0, -2315798183384130627, 1415442519060208520102098146]
c-invariants: [111158312802438270096, -1222942336468020161368212798144]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 636729961016074618212602433984860064248378160
discriminant (Δ): -70657923773953800383052492099299921329642524415200000000
Faltings height: 9.4765
naive height: 138.5576
primes of bad reduction: 2, 3, 5, 137, 269032733, 23993693057739281305146401684243
regulator: 44808456261264.1459138801005183335127817129610410063331179348932280130778
submitted by: Michael Rubinstein
submitted at: 2026-11-03 14:18:29 UTC
last updated: 2026-11-03 14:18:29 UTC

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

(47005479823/9, -9801729695831950/27)
(136035659895, -50170923844711434)
(-1110580492473/841, 1136310910130814750/24389)
(-115044300417/289, 234295372062740250/4913)
(276332956999/2025, 3024976885757888318/91125)
(-697491162657/1369, -2513871374091558750/50653)
(677219868783/11449, -43805654968315973310/1225043)
(766390705917/961, 259389358061692500/29791)
(1686137745813/1849, 625889994996927420/79507)
(993244047, -9754922895750)
(299941752258/289, -55995979846375350/4913)
(733649922, -10551716277750)
(21745253469/25, -961848448217172/125)
(4905832520517/5041, -3269278240139566260/357911)
(1736578222, -51292139560300)
(828149806, -8098544578780)
(370257695216253/310249, -3239666361915744599100/172808693)

Commentary

Mestre ten-root construction, roots 0,3,29,34,45,74,107,124,172,176: 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 14:18:29 UTC · history