curve #6505

y² = x³ + x² − 118921934227546178500x + 702902268232433253444380573984

a-invariants: [0, 1, 0, -118921934227546178500, 702902268232433253444380573984]
c-invariants: [5708252842922216568016, -607307559787071848033478115330240]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 13319113901119342563617347346081167126281944996568
discriminant (Δ): -105800869410754846603700885459678760198441160026722098539341568
Faltings height: 10.6044
naive height: 150.9732
primes of bad reduction: 2, 3, 17, 151, 302322149, 396160157749, 1805084182240184448733271
regulator: 909304895537398.58356067193965252913294086622144373323688229819218279604345
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:54:50 UTC
last updated: 2026-11-04 04:54:50 UTC

Witness: 18 independent points log in to add more points →

(408038755580, 260554881377203872)
(489815234042, -342722068716243906)
(1031328060857/16, 1033654000230093771/64)
(23536250624858/1369, -97957890586064495178/50653)
(1021904380358/169, -994489726707666522/2197)
(2293966366, 664959283070406)
(-111540715207030/12769, 1496147649747107789934/1442897)
(-850298777924/81, -649785476136880576/729)
(-3672254562232/289, -1979627350685739192/4913)
(-6046913362, 1095858813668490)
(-90455107569316/6889, 14343306374417520672/571787)
(86242116385070/24649, 2221899896693657056446/3869893)
(-12025374490, 627692669536578)
(112990641890, 37812754265788422)
(79403965228/9, 15753863414018888/27)
(39215742694697/256, 244979688916077528789/4096)
(-303397862734/529, 10682868644204433786/12167)
(677959272842/121, 613534443072301434/1331)

Commentary

Mestre ten-root construction, roots 0,3,16,37,52,60,113,135,170,174: 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 >= 18: the 18 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 04:54:50 UTC · history