The GIorder function returns the smallest positive integer i such that $n^i=1\mathbf{mod}m$.

If n and m are not coprime, FAIL is returned.

$\mathrm{with}\left(\mathrm{GaussInt}\right)\:$

$\mathrm{GIorder}\left(7+2\mathrm{I}\,9\right)$

$8$

$\mathrm{GIorder}\left(5+8\mathrm{I}\,4+9\mathrm{I}\right)$

$96$

$\mathrm{GIorder}\left(12\,24\right)$

$\mathrm{FAIL}$