curve #6470

y² = x³ − 10455639065155063647x + 9697299690926155655476256130

a-invariants: [0, 0, 0, -10455639065155063647, 9697299690926155655476256130]
c-invariants: [501870675127443055056, -8378466932960198486331485296320]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 196490377341413160535884163143295278957797085473064
discriminant (Δ): 32528677015805314853490471516151335037190364998989091004672
Faltings height: 9.9368
naive height: 142.9946
primes of bad reduction: 2, 3, 13, 17, 18223, 133319, 5082818406145693511698600758193219481
regulator: 14880942986461649.175758522517106470479333431545035722155401035956421504027
submitted by: Michael Rubinstein
submitted at: 2026-11-04 02:10:32 UTC
last updated: 2026-11-04 02:10:32 UTC

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

(90483665277, -27200734802524062)
(1955089820769/25, -2731385044873494522/125)
(-1851232197894/625, 1893168056381110254/15625)
(389786938809/961, 2214007102177261326/29791)
(732391197, 49320726099138)
(8061614410/9, 875109714573070/27)
(28831759747081/11025, 17897401240291914146/1157625)
(3055234568221/729, 3911433546381271858/19683)
(45979877742693/9409, 250450664020326036774/912673)
(-3216414183, 100260315501138)
(-1278397251, 144825674177826)
(-221094219, 109536173805942)
(58689894, 95309293880736)
(3149796681, 89520866607042)
(6724923558, 493473285050904)
(9473404389, 872262105675354)
(16495780633/4, 1532880960496379/8)
(495053357247/121, 250444984066338672/1331)

Commentary

Mestre ten-root construction, roots 0,2,20,31,34,36,63,81,97,116: 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); the sets with S >= 31, or S >= 30 and one bad prime in that range, or S >= 29 and two or more bad primes, were screened by exact conductor and by a GRH explicit-formula upper bound on the rank, and the survivors were certified with ellrank. Rank 18, proved: 18 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-11-04 02:10:32 UTC · history