# Properties

 Label 1.169.aw Base Field $\F_{13^{2}}$ Dimension $1$ Ordinary Yes $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{13^{2}}$ Dimension: $1$ L-polynomial: $1 - 22 x + 169 x^{2}$ Frobenius angles: $\pm0.178912375022$ Angle rank: $1$ (numerical) Number field: $$\Q(\sqrt{-3})$$ Galois group: $C_2$ Jacobians: 8

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 8 curves, and hence is principally polarizable:

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 148 28416 4827316 815766528 137859194068 23298094520064 3937376473771252 665416609532571648 112455406944759851284 19004963774663406302976

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 148 28416 4827316 815766528 137859194068 23298094520064 3937376473771252 665416609532571648 112455406944759851284 19004963774663406302976

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{13^{2}}$
 The endomorphism algebra of this simple isogeny class is $$\Q(\sqrt{-3})$$.
All geometric endomorphisms are defined over $\F_{13^{2}}$.

## 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 1.169.w $2$ (not in LMFDB) 1.169.ab $3$ (not in LMFDB) 1.169.x $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 1.169.w $2$ (not in LMFDB) 1.169.ab $3$ (not in LMFDB) 1.169.x $3$ (not in LMFDB) 1.169.ax $6$ (not in LMFDB) 1.169.b $6$ (not in LMFDB)