Conway and Sloane, as well as Cassels, and also O'Meara, all have their own idiosyncratic way of expressing the following result (for good primes): Every quadratic form with coefficients in $\mathbb{Z}_p$ is equivalent by a change of variables with coefficients $\mathbb{Z}_p$ to one expressible as $\oplus_{i} p^{e_i}h(\epsilon_i, m_i)$, where $h(\epsilon_i, m_i)$ is $y_1^2+y_2^2+\ldots+y_{m_i}^2$ if $\epsilon_i=0$, and $y_1^2+y_2^2+\ldots+ry_{m_i}^2$ with $r$ a quadratic nonresidue if $\epsilon_i=1$.

Furthermore there is a quadratic form $f$ over the integers with given local components $f_i$ if and only if the compatibility conditions for rational equivalence are satisfied.

In the case of rational equivalence this has been given an effective treatment by Kirschmer and a coauthor whose name escapes me right now. However, I don't know of anyone who has implemented (ideally with code that is available) the genus symbol.

Conway and Sloane, as well as Cassels, and also O'Meara, all have their own idiosyncratic way of expressing the following result (for good primes): Every quadratic form with coefficients in $\mathbb{Z}_p$ is equivalent by a change of variables with coefficients $\mathbb{Z}_p$ to one expressible as $\oplus_{i} p^{e_i}h(\epsilon_i, m_i)$, where $h(\epsilon_i, m_i)$ is $y_1^2+y_2^2+\ldots+y_{m_i}^2$ if $\epsilon_i=0$, and $y_1^2+y_2^2+\ldots+ry_{m_i}^2$ with $r$ a quadratic nonresidue if $\epsilon_i=1$.

Furthermore there is a quadratic form $f$ over the integers with given local components $f_i$ if and only if the compatibility conditions for rational equivalence are satisfied.

In the case of rational equivalence this has been given an effective treatment by Kirschmer and a coauthor whose name escapes me right now: given a discriminant and a list of primes where the form should have negative Hasse-Witt invariant, they compute a quadratic form. What I would like is a function that given a discriminant (as an integer), and a list of local components, computes a quadratic form in the given genus, i.e. computes the form $f$ described above.