# Properties

 Label 2.3.ab_b Base Field $\F_{3}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

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## Invariants

 Base field: $\F_{3}$ Dimension: $2$ L-polynomial: $1 - x + x^{2} - 3 x^{3} + 9 x^{4}$ Frobenius angles: $\pm0.201748855633$, $\pm0.672988571819$ Angle rank: $2$ (numerical) Number field: 4.0.16317.1 Galois group: $D_{4}$ Jacobians: 2

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

This isogeny class contains the Jacobians of 2 curves, and hence is principally polarizable:

• $y^2=x^5+x^4+2x+1$
• $y^2=x^6+2x^4+x^3+2x^2+2x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 7 105 553 8925 70672 522585 4970413 43063125 377466187 3477062400

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 11 21 107 288 719 2271 6563 19173 58886

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The endomorphism algebra of this simple isogeny class is 4.0.16317.1.
All geometric endomorphisms are defined over $\F_{3}$.

## 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 2.3.b_b $2$ 2.9.b_n