Let $Q$ be the diagonal quadratic form denoted by $$Q(x_1, \ldots, x_k) = \sum_{i = 1}^{k} d_i x_i^2, \quad x_i \in \mathbb{Q}, \quad d_i > 0$$
Also let ${\mathbb{Q}^{*}}^2$ be the set of nonzero squares of rational numbers, i.e. ${\mathbb{Q}^{*}}^2 = \{q^2 | q \in \mathbb{Q}, q \ne 0\}$
$V = \{ p | p \text{ is prime}\} \cup \{\infty\} $.
Hasse-Witt invariant of the $Q$ for any $\nu \in V$ is denoted as $$E_\nu(Q) = \prod_{i < j} (d_i, d_j)_\nu \in \{\pm 1\},$$ where $(\cdot, \cdot)_\nu$ is the Hilbert symbol.
It is known that $(a, b)_\nu = 1$ for almost all $\nu \in V$ and $\prod_{\nu \in V} (a, b)_\nu = 1$.
Is there any algorithm or criteria to determine if for fixed $k, D, \epsilon_\nu$ there exist $d_1, \ldots, d_k$ such that
- $det(Q) = d_1 d_2 \ldots d_k \equiv D \mod (\mathbb{Q}^{*})^{2}$
- for any $\nu \in V$ holds $E_\nu(Q) = \epsilon_\nu$
