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formula-elliptic-integral: $K(1-m_r)/K(m_r)=\sqrt r$, with $K(m)$ as in HREF{Complete_elliptic_integral_of_the_first_kind_K}[the complete elliptic integral of the first kind]; equivalently the nome is $q=\exp(-\pi- K(1-m_r)/K(m_r))=e^{-\pi\sqrt r}$ CITE{DLMFJacobi}.+ K(1-m_r)/K(m_r))=e^{-\pi\sqrt r}$ CITE{DLMFJacobi}. The values $K(m_r)$ themselves+ are in HREF{T296}[their own table]. formula-complement: $\sqrt{1-k_r^2}=k_{1/r}$, where $k_s=\sqrt{\lambda(i\sqrt s)}$ for any real $s>0$; this is $\lambda(-1/\tau)=1-\lambda(\tau)$ CITE{DLMFModularFunctions}.
of the first kind $K(m)$] relation: $m_r$ is the parameter at which $K(1-m)/K(m)=\sqrt r$+- table: HREF{T296}[Complete elliptic integral of the first kind $K(m_r)$ at the singular+ values]+ relation: holds $K(m_r)$ at the parameters $m_r$ stored here, for the same $r$;+ by CITE{formula-elliptic-integral} it also gives $K(1-m_r)=\sqrt r\,K(m_r)$+- table: HREF{T299}[Values of the elliptic alpha function $\alpha(r)$]+ relation: holds $\alpha(r)$ for the same $r$, built from $K(m_r)$ and $E(m_r)$ at+ the parameters $m_r$ stored here - table: HREF{Rational_singular_moduli}[Rational singular moduli] relation: both evaluate modular functions at imaginary quadratic points; that table
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