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Where can I find some nice formulas for the lambda invariant of an elliptic curve? I vaguely recall there's a nice product formula in terms of $q$, but a Google search didn't give me much. Also, are there other formulas that express the $\lambda$ function in terms of $q$ or give a precise formula for the coefficients of its $q$-expansion?

At the moment I have two books at my disposal: Silverman's Advanced Topics in the Arithmetic of Elliptic Curves, and Diamond and Shurman. Silverman gives nice formulas for the $j$-invariant, but I can't find much on the $\lambda$-invariant, unfortunately.

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    $\begingroup$ Have you seen this? $\endgroup$ – J. M. is a poor mathematician Oct 22 '11 at 12:39
  • $\begingroup$ I have never seen that! Thank you very much!! $\endgroup$ – Bana Oct 22 '11 at 14:25
  • $\begingroup$ Make sure it matches the definition given in your books; I'm not sure what normalization they use there... $\endgroup$ – J. M. is a poor mathematician Oct 22 '11 at 14:26
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Just to settle this: there are a number of formulae for the elliptic lambda function that can be found in the DLMF, the Wolfram Functions site, or MathWorld. (Yes, the modular lambda function and the inverse nome are more or less the same thing.)

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