curve #6510

y² = x³ − 840750115979527767x + 308129759389150392179800890

a-invariants: [0, 0, 0, -840750115979527767, 308129759389150392179800890]
c-invariants: [40356005567017332816, -266224112112225938843347968960]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 164297669584601186159725572720336551472620693277672
discriminant (Δ): -2981016916943003921682060791437786389919229858830080768
Faltings height: 9.2168
naive height: 135.5083
primes of bad reduction: 2, 3, 7, 325987439651986480475645977619715379905993439043
regulator: 2584668277061101.9918829568474191511945768389806982861730441324810772830084
submitted by: Michael Rubinstein
submitted at: 2026-11-04 04:55:26 UTC
last updated: 2026-11-04 04:55:26 UTC

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

(7881983598, -695237245220196)
(1653431526769/576, -2029257369732113801/13824)
(-182513919159/784, 486541944533544507/21952)
(991827785929/3969, 2665514276285853826/250047)
(609647038905/1369, 237786421104407070/50653)
(33745228843/169, -26746919628439724/2197)
(5628224271219/12769, -6947467493398972596/1442897)
(141274960243957/178929, 884344715453202126382/75686967)
(6855173726142/7921, 10661544616265286048/704969)
(21962737377373/40401, -27816078168789271046/8120601)
(-349169823294/361, 100911996829769958/6859)
(561052447, -3610104382108)
(649684078, -6011036426204)
(8395836009/16, -216494047357749/64)
(91729055329473/157609, -249901487094571721766/62570773)
(-44935970221707/192721, -1875641654014085544174/84604519)
(288490737, -9465269676858)
(345856044814/841, -137806777361427184/24389)

Commentary

Mestre ten-root construction, roots 0,4,14,21,37,65,113,141,178,197: 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 04:55:26 UTC · history