A276183 Genus of the quotient of the modular curve X_0(n) by the Fricke involution.
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 2, 1, 1, 1, 2, 0, 1, 0, 0, 1, 2, 1, 1, 1, 1, 2, 3, 0, 3, 1, 2, 1, 1, 1, 3, 2, 2, 2, 4, 0, 2, 2, 2, 1, 3, 2, 5, 1, 2, 1, 4, 1, 4, 3, 3, 2, 4, 1, 4, 2, 4, 4, 4, 1, 3, 3, 2, 3, 3, 1, 7
Offset: 1
Keywords
Examples
G.f. = x^22 + x^28 + x^30 + x^33 + x^34 + x^37 + x^38 + x^40 + 2*x^42 + x^43 + x^44 + ...
Links
- Gheorghe Coserea, Table of n, a(n) for n = 1..54321
- Harvey Cohn, Fricke's Two-Valued Modular Equations, Math. Comp. 51 (1988), 787-807.
- Harvey Cohn, A Numerical Survey of the Reduction of Modular Curve Genus by Fricke's Involutions, Number Theory (New York Seminar 1989-1990), p. 100.
- Fell, Harriet; Newman, Morris; Ordman, Edward; Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
- Andrew P. Ogg, Automorphismes de courbes modulaires, Séminaire Delange-Pisot-Poitou. Théorie des nombres, vol. 16, no. 1 (1974-1975), talk no. 7, p. 1.
Programs
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Mathematica
f[n_] := If[n < 1, 0, 1 + Sum[MoebiusMu[d]^2 n/d/12 - EulerPhi[GCD[d, n/d]]/2, {d, Divisors@ n}] - Count[(#^2 - # + 1)/n & /@ Range@ n, ?IntegerQ]/3 - Count[(#^2 + 1)/n & /@ Range@ n, ?IntegerQ]/4]; g[n_] := Ceiling[k0 = k /. FindRoot[EllipticK[1 - k^2]/EllipticK[k^2] == Sqrt@ n, {k, 1/2, 10^-10, 1}, WorkingPrecision -> 600, MaxIterations -> 100]; Exponent[MinimalPolynomial[RootApproximant[k0^2, 24], x], x]/2]; r[n_] := If[MemberQ[{3, 7}, #], 3 + (# - 1)/2, 3] &@ Mod[n, 8]; a[n_] := If[n <= 4, 0, (1 + f@ n)/2 - r[n] g[n]/12]; Table[Print["a(", n, ") = ", an = a[n]]; an, {n, 102}] (* Michael De Vlieger, Oct 28 2016, after Michael Somos at A001617 and Jean-François Alcover at A000003 *) ClassList[n_?Negative] := Select[Flatten[#, 1] &@Table[ {i, j, (j^2 - n)/(4 i)}, {i, Sqrt[-n/3]}, {j, 1 - i, i}], Mod[#3, 1] == 0 && #3 >= # && GCD[##] == 1 && ! (# == #3 && #2 < 0) & @@ # &] A001617[n_] := If[n < 1, 0, 1 + Sum[MoebiusMu[d]^2 n/d/12 - EulerPhi[GCD[d, n/d]]/2, {d, Divisors@n}] - Count[(#^2 - # + 1)/n & /@ Range[n], _?IntegerQ]/3 - Count[(#^2 + 1)/n & /@ Range[n], _?IntegerQ]/4]; a[n_] := If[0 <= n <= 4, 0, (A001617[n] + 1)/2 - If[Mod[n, 8] == 3, 4, If[Mod[n, 8] == 7, 6, 3]] Length[ClassList[-4 n]]/12] (* David Jao, Sep 07 2020 *) -
PARI
A000003(n) = qfbclassno(-4*n); A000089(n) = { if (n%4 == 0 || n%4 == 3, return(0)); if (n%2 == 0, n \= 2); my(f = factor(n), fsz = matsize(f)[1]); prod(k = 1, fsz, if (f[k, 1] % 4 == 3, 0, 2)); }; A000086(n) = { if (n%9 == 0 || n%3 == 2, return(0)); if (n%3 == 0, n \= 3); my(f = factor(n), fsz = matsize(f)[1]); prod(k = 1, fsz, if (f[k, 1] % 3 == 2, 0, 2)); }; A001615(n) = { my(f = factor(n), fsz = matsize(f)[1], g = prod(k=1, fsz, (f[k, 1]+1)), h = prod(k=1, fsz, f[k, 1])); return((n*g)\h); }; A001616(n) = { my(f = factor(n), fsz = matsize(f)[1]); prod(k = 1, fsz, f[k, 1]^(f[k, 2]\2) + f[k, 1]^((f[k, 2]-1)\2)); }; A001617(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2; a(n) = { my(r = if (n%8 == 3, 4, n%8 == 7, 6, 3)); if (n < 5, 0, (1 + A001617(n))/2 - r * A000003(n)/12); }; vector(102, n, a(n))
Formula
Extensions
New name from David Jao, Sep 07 2020
Comments