# Properties

 Label 1.11.a Base field $\F_{11}$ Dimension $1$ $p$-rank $0$ Ordinary No Supersingular Yes Simple Yes Geometrically simple Yes Primitive Yes Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{11}$ Dimension: $1$ L-polynomial: $1 + 11 x^{2}$ Frobenius angles: $\pm0.5$ Angle rank: $0$ (numerical) Number field: $$\Q(\sqrt{-11})$$ Galois group: $C_2$ Jacobians: 4

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 4 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 12 144 1332 14400 161052 1774224 19487172 214329600 2357947692 25937746704

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 12 144 1332 14400 161052 1774224 19487172 214329600 2357947692 25937746704

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{11}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-11})$$.
Endomorphism algebra over $\overline{\F}_{11}$
 The base change of $A$ to $\F_{11^{2}}$ is the simple isogeny class 1.121.w and its endomorphism algebra is the quaternion algebra over $$\Q$$ ramified at $11$ and $\infty$.
All geometric endomorphisms are defined over $\F_{11^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

This isogeny class has no twists.