# Properties

 Label 1.331.a Base Field $\F_{331}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{331}$ Dimension: $1$ L-polynomial: $1 + 331 x^{2}$ Frobenius angles: $\pm0.5$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-331})$$ Galois group: $C_2$ Jacobians: 12

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is supersingular. $p$-rank: $0$ Slopes: $[1/2, 1/2]$

## Point counts

This isogeny class contains the Jacobians of 12 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 332 110224 36264692 12003393600 3973195810652 1315127885854864 435307306210734212 144086718331745798400 47692703775754250891372 15786284949782603436665104

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 332 110224 36264692 12003393600 3973195810652 1315127885854864 435307306210734212 144086718331745798400 47692703775754250891372 15786284949782603436665104

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{331}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-331})$$.
Endomorphism algebra over $\overline{\F}_{331}$
 The base change of $A$ to $\F_{331^{2}}$ is the simple isogeny class 1.109561.zm and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $331$ and $\infty$.
All geometric endomorphisms are defined over $\F_{331^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

This isogeny class has no twists.