As with any abelian variety, the Jacobian $J_H$ of a modular curve $X_H$ can be decomposed into simple isogeny factors. For modular curves, these simple isogeny factors are modular abelian varieties corresponding to Galois orbits of weight 2 newforms.

We list two types of information about the **isogeny decomposition** of $J_H$:

- the multiset of dimensions of the simple isogeny factors: $1^3\cdot 2^2$ denotes $3$ factors of dimension $1$ and $2$ factors of dimension $2$;
- the multiset of newforms corresponding to the modular abelian varieties in the isogeny decomposition, listed by label: $\texttt{11.2.a.a}^3\cdot \texttt{13.2.e.a}^2$ denotes five simple isogeny factors, three isogenous to the modular abelian variety corresponding to the newform labelled $\texttt{11.2.a.a}$ and two isogenous to the modular abelian variety corresponding to the newform labelled $\texttt{13.2.3.a}$.

When $X_H$ has genus zero, $J_H$ is the trivial abelian variety of dimension zero, and no isogeny decomposition information is listed.

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**Knowl status:**

- Review status: beta
- Last edited by Bjorn Poonen on 2022-03-24 18:18:04

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**History:**(expand/hide all)

- 2022-03-24 18:18:04 by Bjorn Poonen
- 2022-03-20 20:29:17 by Andrew Sutherland
- 2022-03-20 20:28:49 by Andrew Sutherland
- 2022-03-20 20:27:01 by Andrew Sutherland

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