A294705 Order of the general symplectic group of 4 X 4 matrices over Z_n.

1, 720, 103680, 1474560, 37440000, 74649600, 1659571200, 3019898880, 18366600960, 26956800000, 257213088000, 152882380800, 1644455554560, 1194891264000, 3881779200000, 6184752906240, 32143905423360, 13223952691200, 110052644025600, 55207526400000, 172064342016000, 185193423360000

Offset

1,2

Comments

Let M be any fixed nonsingular skew-symmetric 4 X 4 matrix over the integers mod n. Then a(n) is the number of invertible 4 X 4 matrices A over the integers mod n such that A^T * M * A = c*M for some nonzero constant c (mod n), where A^T denotes the transpose of A.

Links

Table of n, a(n) for n=1..22.

Hanson Smith, Frobenius finds non-monogenic division fields of abelian varieties, Int. J. Number Theory 18 (2022), no. 10, 2299-2315.

Formula

a(n) = Product_{primes p dividing n} p^(11*v_p(n) - 7)*(p - 1)*(p^2 - 1)*(p^4 - 1), where v_p(n) is the largest power k such that p^k divides n.

Sum_{k=1..n} a(k) ~ c * n^12 / 12, where c = Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4 - 1/p^5 + 1/p^6 + 1/p^7 - 1/p^8) = 0.5251079212... . - Amiram Eldar, Aug 07 2023

Mathematica

f[p_, e_] := p^(11*e - 7)*(p - 1)*(p^2 - 1)*(p^4 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 25] (* Amiram Eldar, Aug 07 2023 *)

Prog

(Sage)
def a(n):
    return product([p^(11*n.valuation(p) - 7)*(p - 1)*(p^2 - 1)*(p^4 - 1)
        for p in n.prime_factors()])

Crossrefs

Cf. A305186.

Sequence in context: A364771 A003931 A158049 * A222004 A047803 A166765

Adjacent sequences: A294702 A294703 A294704 * A294706 A294707 A294708

Keyword

nonn,mult

Author

Robin Visser, Aug 06 2023

Status

approved