A365138 Genus of the quotient of the modular curve X_1(n) by the Fricke involution.
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 1, 3, 1, 2, 3, 5, 2, 6, 4, 6, 5, 10, 4, 12, 8, 10, 10, 11, 8, 20, 13, 15, 12, 24, 12, 28, 17, 20, 22, 33, 18, 34, 23, 31, 27, 45, 25, 39, 29, 42, 39, 56, 28, 62, 44, 47, 46, 59, 39, 77, 51, 65, 48, 85, 48, 93, 66, 71, 67, 89, 60, 109
Offset: 1
Keywords
Links
- C. H. Kim and J. K. Koo, Estimation of Genus for Certain Arithmetic Groups, Communications in Algebra, 32:7 (2004), 2479-2495.
Programs
-
Mathematica
A000003[n_] := Length[Select[ Flatten[#, 1] &@ Table[{i, j, (j^2 + 4 n)/(4 i)}, {i, Sqrt[4 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]; A029937[n_] = If[n < 5, 0, 1 + Sum[d^2*MoebiusMu[n/d]/24 - EulerPhi[d]*EulerPhi[n/d]/4, {d, Divisors[n]}]]; A276183[n_] := If[0 <= n <= 4, 0, (A001617[n] + 1)/2 - If[Mod[n, 8] == 3, 4, If[Mod[n, 8] == 7, 6, 3]]*A000003[n]/12]; A365138[n_] := (A029937[n] - A001617[n])/2 + A276183[n]