There is now also a Magma implementation of this algorithm, implemented by
John Cremona. His Magma implementation contains a feature that is not yet
present in the Maple implementation given here, namely the following:

Suppose a, b, c in Z[t] are squarefree and have gcd 1 pairwise.
If any of the factors of a,b,c is linear, then it is not necessary
to solve a conic over Z    (the Maple implementation given here
only avoids a conic over Z if the degrees of a,b,c do not all have
the same parity).