curve #6109

y² + xy + y = x³ − x² − 92608886111987607x + 10432304298343322885158575

a-invariants: [1, -1, 1, -92608886111987607, 10432304298343322885158575]
c-invariants: [4445226533375405121, -9013510893765111572587685985]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 201612979168923207004963766111743393408646
discriminant (Δ): 3816244637494318406672231279804842579593723788787712
Faltings height: 8.6611
naive height: 128.8151
primes of bad reduction: 2, 7, 11, 31, 42231457722857814621902757878454837329
regulator: 40727474448249963.23102231356193332333669976788330610868288011668481083
submitted by: Michael Rubinstein
submitted at: 2026-10-29 02:29:54 UTC
last updated: 2026-10-29 02:29:54 UTC

Witness: 16 independent points log in to add more points →

(-62983737, 4001944111562)
(258895381, 1951594163872)
(-46587839153837/1739761, 8239733690641210664466/2294744759)
(-4914648172965/17689, 9025190683221763590/2352637)
(-1864106988677/5329, 60555895170184654/389017)
(-308072087005/9409, 3344613276463858270/912673)
(-14231438093/49, 1228633858187826/343)
(927064639/9, 38047343304602/27)
(4393830547/49, 578941298516946/343)
(37190323323/169, 1850415261704534/2197)
(201534159859/289, 82935019973891598/4913)
(237389119819/121, 114341769512822810/1331)
(3934253068171/1681, 7740444687262409486/68921)
(12224340193171/57121, 8789955934756082834/13651919)
(12316457697707/85849, 7910625327972542918/25153757)
(444554121697171/2187441, 235215936812979430994/3235225239)

Commentary

Mestre ten-root construction, roots 0,10,54,60,61,66,67,74,81,103: 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 16, proved: 16 independent points listed, matching the 2-descent upper bound. Torsion trivial.

last edited by Michael Rubinstein at 2026-10-29 02:29:54 UTC · history