  In this paper Seidel matrices are studied, and their spectrum and several
related algebraic properties are determined for order $n\leq 13$. Based on this
Seidel matrices with exactly three distinct eigenvalues of order $n\leq 23$ are
classified. One consequence of the computational results is that the maximum
number of equiangular lines in $\mathbb{R}^{12}$ with common angle $1/5$ is
exactly $20$.
