I would like to know what algorithms there are to compute the linearly independent generators $(1,i,j,k)$ for quaternion algebra containing the endomorphism ring of a supersingular curve. The curve in question is pretty small (around 16 bits, but I have no idea how to perform a brute force search).

One has that the Frobenius endomorphism would give us a generator linearly independent to the integers (that gives us $i^2=-p$), so how can one find the other? I have tried to look at Hilbert polynomials $h_D(X)$ for different $D$'s to see if $j_E$, the $j$-invariant of the elliptic curve, is a root, but there are more than one solution, so could one take the smallest $D$ so that $j^2=-D$?

Also, once the first part is completed, how does one find the $\mathbb{Z}$-basis that generates the endomorphism ring?

Lastly, how does one check if the endomorphism ring or the quaternion algebra obtained from the algorithms are correct?

(Aside: Does anyone have a reference to show that roots of $D$-Hilbert polynomials are $j$-invariants of elliptic curves with endomorphism ring containing the imaginary quadratic field with discriminant $D$? I've seen it used all the time, but have not seen an explicit reference.)

Edit: I have found a paper by Pizer that contains a proposition (5.1) that describes the quaternion algebra the endomorphism ring lives in. Another proposition (5.2) describes a maximal order, but this still does not help my cause.