A029937 Genus of modular curve X_1(n).

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 1, 2, 5, 2, 7, 3, 5, 6, 12, 5, 12, 10, 13, 10, 22, 9, 26, 17, 21, 21, 25, 17, 40, 28, 33, 25, 51, 25, 57, 36, 41, 45, 70, 37, 69, 48, 65, 55, 92, 52, 81, 61, 85, 78, 117, 57, 126, 91, 97

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Also the dimension of the space of cusp forms of weight two on Gamma1(n). [Steven Finch, Apr 03 2009]

References

• F. Hirzebruch et al., Manifolds and Modular Forms, Vieweg, 2nd ed. 1994, p. 161.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• S. R. Finch, Modular forms on SL_2(Z), December 28, 2005. [Cached copy, with permission of the author]
• Chang Heon Kim, Ja Kyung Koo, On the genus of some modular curves of level N, Bull Austral. Math. Soc. 54 (1996) 291-297.
• A. V. Sutherland, Torsion subgroups of elliptic curves over number fields, 2012. - From _N. J. A. Sloane_, Feb 03 2013

Crossrefs

Cf. A001617, A029938. [Steven Finch, Apr 03 2009]

Programs

Maple

with(numtheory); A029937 := proc(n) local i,j; j := 1+(1/24)*phi(n)*A001615(n); for i in divisors(n) do j := j-(1/4)*phi(i)*phi(n/i) od; j; end;

Mathematica

a[n_ /; n<5] = 0; a[n_] := 1+Sum[d^2*MoebiusMu[n/d]/24 - EulerPhi[d]*EulerPhi[n/d]/4, {d, Divisors[n]}]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 13 2014 *)

PARI

A029935(n) = {
  my(f = factor(n), fsz = matsize(f)[1],
     g = prod(k=1, fsz, f[k,1]),
     h = prod(k=1, fsz, sqr(f[k,1]-1)*f[k,2] + sqr(f[k,1])-1));
  return(h*n\sqr(g));
};
a(n) = {
  if (n < 5, return(0));
  my(f = factor(n), fsz = matsize(f)[1],
     g = prod(k=1, fsz, f[k,1]),
     h = prod(k=1, fsz, sqr(f[k,1]) - 1));
  return(1 + sqr(n\g)*h/24 - A029935(n)/4);
};
vector(63, n, a(n))  \\ Gheorghe Coserea, Oct 23 2016

Formula

a(n) = 1 + A115000(n) - A029935(n)/4, n > 4. [Kim and Koo, Theorem 1]