in , , known as the modular equation, whose n + 1 roots in are given by: and where is 1 or −1 depending on whether 2 is a quadratic residue modulo n or not, respectively, and . For n = 5, we have the modular equation: with six roots in as shown above. The modular equation with n = 5 may be related to the Bring–Jerrard quintic by the following function of the six roots of the modular equation (In Hermite's Sur la théorie des équations modulaires et la résolution de l'équation du cinquième degré, the first factor is incorrectly given as ): Alternatively, the formula is useful for numerical evaluation of . According to Hermite, the coefficient of in the expansion is zero for every . The five quantities , , , , are the roots of a quintic equation with coefficients rational in : which may be readily converted into the Bring–Jerrard form by the substitution: leading to the Bring–Jerrard quintic: where The Hermite–Kronecker–Brioschi method then amounts to finding a value for that corresponds to the value of , and then using that value of to obtain the roots of the corresponding modular equation. We can use root finding algorithms to find from the equation (i.e. compute a partial inverse of ). The roots of the Bring–Jerrard quintic are then given by: for . An alternative, "integral", approach is the following: Consider where Then is a solution of where The roots of the equation are: where (note that some important references erroneously give it as ). One of these roots may be used as the elliptic modulus . The roots of the Bring–Jerrard quintic are then given by: for . It may be seen that this process uses a generalization of the nth root, which may be expressed as: or more to the point, as The Hermite–Kronecker–Brioschi method essentially replaces the exponential by an "elliptic transcendent", and the integral (or the inverse of on the real line) by an elliptic integral (or by a partial inverse of an "elliptic transcendent"). Kronecker thought that this generalization was a special case of a still more general theorem, which would be applicable to equations of arbitrarily high degree. This theorem, known as Thomae's formula, was fully expressed by Hiroshi Umemura in 1984, who used Siegel modular forms in place of the exponential/elliptic transcendents, and replaced the integral by a hyperelliptic