# Properties

 Label 2.7.ae_l 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: $1 - 4 x + 11 x^{2} - 28 x^{3} + 49 x^{4}$ Frobenius angles: $\pm0.158901191781$, $\pm0.538942184569$ Angle rank: $2$ (numerical) Number field: 4.0.138768.1 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 29 2697 112868 5655609 288523349 13981636368 678495801173 33235712260137 1628982534424868 79792878597781497

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 4 56 328 2356 17164 118838 823876 5765284 40367704 282477416

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{7}$
 The endomorphism algebra of this simple isogeny class is 4.0.138768.1.
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.e_l $2$ 2.49.g_af