Endomorphism ring notation (awaiting review)

In the graph, each vertex correspond to an endomorphism of abelian varieties in the isogeny class and each edge correspond to an inclusion.

For each endomorphism ring $S$, we use the following notation $[\mathrm{index}]^{\#\mathrm{weak}\cdot \#\mathrm{Pic}}_i$ :

• $\mathrm{index}$ is the index of the inclusion (of additive groups) $S\subseteq \mathcal{O}$, where $\mathcal{O}$ is the maximal order of the étale $\mathbb{Q}$-algebra $\mathbb{Q}[F]$;

• $i$ is a counter that uniquely identifies the order $S$ among all the overorders of $\mathbb{Z}[F,V]$ with index in $\mathcal{O}$ equal to $\mathrm{index}$. See also isogeny label.

• $\#\mathrm{weak}$ is the number of weak equivalence classes represented by fractional $\mathbb{Z}[F,V]$-ideals $I$ with multiplicator ring equal to $S$, that is, satisfying $(I:I)=S$; if $S$ is Gorenstein, that is, $\#\mathrm{weak} = 1 $ this number is omitted from the notation.

• $\#\mathrm{Pic}$ is the size of the Picard groups of $S$, that is, the number of invertible fractional $S$-ideals.