# Properties

 Label 1.13.ac Base Field $\F_{13}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{13}$ Dimension: $1$ L-polynomial: $1 - 2 x + 13 x^{2}$ Frobenius angles: $\pm0.410543812489$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3})$$ Galois group: $C_2$ Jacobians: 4

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $1$ Slopes: $[0, 1]$

## 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 $A(\F_{q^r})$ 12 192 2268 28416 370092 4826304 62763132 815766528 10604381004 137857789632

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 12 192 2268 28416 370092 4826304 62763132 815766528 10604381004 137857789632

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3})$$.
All geometric endomorphisms are defined over $\F_{13}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 1.13.c $2$ 1.169.w 1.13.af $3$ (not in LMFDB) 1.13.h $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.13.c $2$ 1.169.w 1.13.af $3$ (not in LMFDB) 1.13.h $3$ (not in LMFDB) 1.13.ah $6$ (not in LMFDB) 1.13.f $6$ (not in LMFDB)