curve #6122

y² + xy + y = x³ − x² − 15793126247508684444x + 24154450924865994000420817119

a-invariants: [1, -1, 1, -15793126247508684444, 24154450924865994000420817119]
c-invariants: [758070059880416853297, -20869445595672903546901710151209]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 133189757005333373984927132272779327740156243097818
discriminant (Δ): 61646612650404541482175553318191366285875836805338865664
Faltings height: 9.7213
naive height: 144.2319
primes of bad reduction: 2, 113, 2411, 151830629, 2226833419654303, 6397931882288571973
regulator: 119652716301597166823793.46205957175763553463496204566753775219499364698818
submitted by: Michael Rubinstein
submitted at: 2026-10-29 03:14:15 UTC
last updated: 2026-10-29 03:14:15 UTC

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

(-841179993472485683/1452905689, 10076211657978856820928856785/55380406147613)
(-886571823533339/388129, 53148381012832838440203/241804367)
(259179362888893/124609, 766674502563305398829/43986977)
(508900288050637/26569, 11250481503440281075287/4330747)
(1696360727028281/942841, 36201074214254286222115/915498611)
(2356963826795821/177241, 109811694388565628570993/74618461)
(7912830677411581/3367225, 26529621699080357959199/6178857875)
(26419290284336261/11621281, 11984689609144941833845/39616946929)
(218962054867662509/94109401, 1878867607564552633628209/912955299101)
(412342250682365621/164583241, 37336777518301774342514857/2111438398789)
(4305434285939492719/2060615236, 1558905500826110142742714179/93539568022984)
(3733017571652531097/3101710249, 14338842936139142028442021265/172743548897557)
(6096851347162197619/2936447721, 2820625929013876923403453139/159123165553269)
(48420400931152622917/76862572081, 2562042440723411449339461227885/21309456346308521)
(98789146619795703669/46098237025, 121762009141530903318592457493/9897521980452625)
(312452527460365366089/134871828001, 33603801845612191063644357321/49531543961539249)
(187459415429267966880853/82836737957961, 1471216497875223896817275790467945/753936002585505344709)
(40023184945023819725569149/17287317786750025, 317660926137724701606365126063444217/2272958065312537790781125)

Commentary

Mestre ten-root construction, roots 0,3,5,19,44,45,68,83,90,113: 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 2026-10-29 03:14:15 UTC · history