A118778 Total degree of the classical modular curve X_n(0). Also the degree of the classical modular polynomial.

1, 4, 6, 9, 10, 18, 14, 20, 20, 30, 22, 38, 26, 42, 40, 42, 34, 62, 38, 60, 56, 66, 46, 82, 54, 78, 66, 84, 58, 122, 62, 88, 88, 102, 84, 126, 74, 114, 104, 126, 82, 168, 86, 132, 128, 138, 94, 172, 104, 166, 136, 156, 106, 198, 132, 170, 152, 174, 118, 254, 122, 186, 172

Offset: 1

Author

Gene Ward Smith, May 22 2006

Keywords

• nice
• nonn

Comments

This is the total degree of the classical modular curve relating j(z) to j(nz), where j is the j-invariant, or elliptic modular function. If F_n(x, y) = 0 is the equation for the curve (the classical modular equation) then F_n(x, x) is the classical modular polynomial and the sequence is also the sequence of degrees for it. When n is a prime, the degree is 2n.

References

• Serge Lang, ''Elliptic Functions'', Addison-Wesley, 1973.

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Programs

Maple

with(numtheory): degx := proc (n) # degree of the classical modular curve X0(n) local a, s; s := 0; for a in divisors(n) do if a^2 > n then s := s + 2*a*phi(igcd(a, n/a))/igcd(a, n/a) fi od; if issqr(n) then s := s+phi(sqrt(n)) fi; s end:

Mathematica

degx[n_] := Module[{s = 0}, Do[ If[ a^2 > n, s = s + 2*a*EulerPhi[ GCD[a, n/a]] / GCD[a, n/a]], {a, Divisors[n]}]; If[ IntegerQ[ Sqrt[n]], s = s + EulerPhi[ Sqrt[n] ] ]; s]; Table[ degx[n], {n, 1, 63}] (* Jean-François Alcover, Jan 29 2013, translated from Maple *)

PARI

a(n)=2*sumdiv(n,d,if(d^2>n, my(g=gcd(d,n/d)); d*eulerphi(g)/g)) + if(issquare(n,&n),eulerphi(n)) \\ Charles R Greathouse IV, Jan 29 2013