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
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
Programs
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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;
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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
Comments