Enumeration of Seidel matrices

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