# Properties

 Label 2.7.af_q Base Field $\F_{7}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{7}$ Dimension: $2$ L-polynomial: $1 - 5 x + 16 x^{2} - 35 x^{3} + 49 x^{4}$ Frobenius angles: $\pm0.169178782589$, $\pm0.473594973839$ Angle rank: $2$ (numerical) Number field: 4.0.57800.1 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 26 2756 121784 5666336 284585886 13993468736 680297445854 33223427868800 1628029116513176 79793296617432996

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 57 354 2361 16933 118938 826059 5763153 40344078 282478897

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
 The endomorphism algebra of this simple isogeny class is 4.0.57800.1.
All geometric endomorphisms are defined over $\F_{7}$.

## 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 2.7.f_q $2$ 2.49.h_e