curve #6447

y² + xy = x³ − 192757859481299043905x + 1018859061105647169308305679241

a-invariants: [1, 0, 0, -192757859481299043905, 1018859061105647169308305679241]
c-invariants: [9252377255102354107441, -880294228809157720165029638025385]
rank (lower bound): ≥ 17
torsion subgroup: trivial
conductor (N): 476374753687193055372372517588509157743651102
discriminant (Δ): 9922201523464135751507534061798767835849696838714640881631232
Faltings height: 10.5137
naive height: 151.7375
primes of bad reduction: 2, 3, 11, 37, 43, 61, 29243463533, 4904099656411, 518581108686019
regulator: 11183103279591.0673230806933802363925841887727436024687034576916198965504
submitted by: Michael Rubinstein
submitted at: 2026-11-03 14:23:40 UTC
last updated: 2026-11-03 14:23:40 UTC

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

(-13965478556, 993517632802561)
(80332381208146/25, -719999234472481896721/125)
(-1548640268534/529, -15187137400261677523/12167)
(11259318418690/841, -22324649284198953049/24389)
(135836333360374/15625, 23721609110775983807/1953125)
(103689937844266/14161, -8224910664220970131/1685159)
(732270732571/100, 2859101764384351/1000)
(239853304949938/25281, 843763042434648326477/4019679)
(311911095720970/32041, 1461375065488947706841/5735339)
(3726173298648586/609961, 125403903066123323227111/476379541)
(339887184228/49, 42764446056223767/343)
(-12098410223066/3025, 218567596293608189413/166375)
(628291779950781706/98465929, 215621760112241652083985017/977077413467)
(22010447144284/61009, 14682742004693285889133/15069223)
(631603744348/121, 523867367174455643/1331)
(3191471134, 660441140356675)
(481233001786/361, -5996320484797626023/6859)

Commentary

Mestre ten-root construction, roots 0,1,23,29,100,119,122,159,179,198: 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 14:23:40 UTC · history