curve #6494

y² + xy + y = x³ − x² − 2341813627836153331412x + 43929609726043057225931293594295

a-invariants: [1, -1, 1, -2341813627836153331412, 43929609726043057225931293594295]
c-invariants: [112407054136135359907761, -37955182802795369699592028545886185]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 275993669292054687436741062820989156987142998942730458
discriminant (Δ): -11744155107108712633724189089819326917844926506042596750627176448
Faltings height: 11.1205
naive height: 159.2434
primes of bad reduction: 2, 3, 167, 10600957, 8660940822751677874735696528448967274511599
regulator: 117240687046943376.4442596274355971032016844565085835990997415473934405642927
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:53:32 UTC
last updated: 2026-11-04 04:53:32 UTC

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

(3203111169537/121, -929642101086481885/1331)
(1340195381925, -1550504484546381971)
(86100285853701/3481, 219142733553900070943/205379)
(2615805162339/100, 755505988416635293/1000)
(238029209287533/6889, 1180828690832860137455/571787)
(117055048942917/2401, 794769716817420480445/117649)
(31381270296621/361, 152948427865874864623/6859)
(16253795518957/169, 58471556759498204529/2197)
(679236030309069/4489, 16890624044366228987071/300763)
(115271159377941/5041, -547118500435548348469/357911)
(-2480920210011/49, -1961560547031382037/343)
(-7575457608387/1849, -581300846735588130281/79507)
(13058321079925/529, -13125679451459246085/12167)
(906879861153/361, -42316490578498822517/6859)
(5804598909165/169, -4402128522359394575/2197)
(5754945252032205/223729, -89258120783641877211667/105823817)
(3338659042173/121, -753620922519581641/1331)
(1059634744793317/83521, 97334528431199143090125/24137569)
(-8889212627283/169, -10206901712251439567/2197)

Commentary

Mestre ten-root construction, roots 0,33,59,60,83,98,114,117,134,186: 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 04:53:32 UTC · history