I am interested in finding the derivation of the duplication, triplication and quintuplication formulae for Klein’s j-invariant, which are equations (13) – (24) of the corresponding page (Klein’s j-invariant) of MathWorld:

http://mathworld.wolfram.com/KleinsAbsoluteInvariant.html

Yes, I've asked the MathWorld team (without luck).

I'm guessing this is a easy, standard result for those who are in the know, apologies for that.

I have looked at Apostol, ‘Modular Functions and Dirichlet Series in Number Theory’ (the first reference on the MathWorld page) and Thm 4.11 on page 89 comes close but doesn't seem to do it for me.

I would be very grateful if someone could give me the reference for the derivation, in the least general case (as I am a bit of a novice in this area).

Best, m

PS The duplication formula is
$J(\tau)=f(t)$ and $J(2\tau)=f(1/t)$
with
$t=\frac{1}{64}\left[\frac{\eta(\tau)}{\eta(2\tau)}\right]^{24}$,

$f(u)=\frac{(u+4)^3}{27u^2}$ and $\eta(z)$ the Dedekind eta.