# Properties

 Label 1.169.aba Base Field $\F_{13^{2}}$ Dimension $1$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{13^{2}}$ Dimension: $1$ L-polynomial: $( 1 - 13 x )^{2}$ Frobenius angles: $0$, $0$ Angle rank: $0$ (numerical) Number field: $$\Q$$ Galois group: Trivial Jacobians: 1

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 1 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 144 28224 4822416 815673600 137857749264 23298075468864 3937376260202256 665416607551718400 112455406930748394384 19004963774605082455104

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 144 28224 4822416 815673600 137857749264 23298075468864 3937376260202256 665416607551718400 112455406930748394384 19004963774605082455104

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The endomorphism algebra of this simple isogeny class is the quaternion algebra over $$\Q$$ ramified at $13$ and $\infty$.
All geometric endomorphisms are defined over $\F_{13^{2}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.169.ba $2$ (not in LMFDB)