Given the Nome q, , entering the definition of Jacobi Theta functions, for instance

FunctionAdvisor(definition, JacobiTheta1)[1];

EllipticModulus computes the corresponding Modulus k,  entering the definition of related elliptic integrals and JacobiPQ elliptic functions.

FunctionAdvisor(definition, EllipticF)[1];

FunctionAdvisor(definition, JacobiSN)[1];

FunctionAdvisor(definition, JacobiAM);

Alternatively, given the Modulus k,  entering Elliptic integrals and JacobiPQ functions, it is possible to compute the corresponding Nome q, , using EllipticNome, which is the inverse function of EllipticModulus.

EllipticModulus is defined in terms of JacobiTheta functions by:

FunctionAdvisor( definition, EllipticModulus );

The JacobiPQ functions can be expressed in terms of JacobiTheta functions using EllipticNome

JacobiSN(z,k) = (1/(k^2))^(1/4) * JacobiTheta1(1/2*Pi*z/EllipticK(k),EllipticNome(k)) / JacobiTheta4(1/2*Pi*z/EllipticK(k),EllipticNome(k));

Alternative popular notations for elliptic integrals and JacobiPQ functions involve a parameter m or a modular angle alpha, as for instance in the Handbook of Mathematical Functions edited by Abramowitz and Stegun (A&S). These are related to k by  and sin(alpha) = k. For example, the Elliptic  function shown in A&S is numerically equal to the Maple  command.