I learned form this question and this answer that Schur index in GAP can be found using LoadPackage("wedderga") the functions "SchurIndex".

But I am working on the field $K=\mathbb Q (\sqrt{-39})$ and with the generalised quaternion group $Q=Q_{16}$ which can be considered to be generated by two matrices over $K$, details are given in this paper. In particular I am interested in the groups given by the wreath products $W(s)=W(s-1)\wr \langle (1,2) \rangle$ for $s\ge 3$ with $W(2)=Q$.

I am wondering how to calculate the Schur index of $Q$ or $W(s)$ for different $s$. Using the "SchurIndex" function in GAP gives me the error "fail: Quaternion Algebra Over NonRational Field, use another method."

From the GAP website, I found this : "the quaternion algebra can be converted to a cyclic algebra and the Schur index of the cyclic algebra can be determined through the solution of norm equations. Currently this functionality is not implemented in GAP, but available in number theory packages such as PARI/GP."

I am not very much familiar with what this suggests and I will very much like to know if there is any way to compute the Schur index by computer or in any other way.

Thanks in advance.


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