# Properties

 Label 2.2.ab_c Base field $\F_{2}$ Dimension $2$ $p$-rank $1$ Ordinary No Supersingular No Simple No Geometrically simple No Primitive Yes Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{2}$ Dimension: $2$ L-polynomial: $( 1 - 2 x + 2 x^{2} )( 1 + x + 2 x^{2} )$ Frobenius angles: $\pm0.250000000000$, $\pm0.615026728081$ Angle rank: $1$ (numerical) Jacobians: 1

This isogeny class is not simple.

## Newton polygon

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

## Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

• $y^2+xy=x^5+x^3+x$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4 40 52 400 1804 3640 13108 64800 244348 992200

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 2 8 8 24 52 56 100 256 476 968

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ac $\times$ 1.2.b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{2}$
 The base change of $A$ to $\F_{2^{4}}$ is 1.16.ab $\times$ 1.16.i. The endomorphism algebra for each factor is: 1.16.ab : $$\Q(\sqrt{-7})$$. 1.16.i : the quaternion algebra over $$\Q$$ ramified at $2$ and $\infty$.
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{2^{2}}$  The base change of $A$ to $\F_{2^{2}}$ is 1.4.a $\times$ 1.4.d. The endomorphism algebra for each factor is:

## 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 2.2.ad_g $2$ 2.4.d_i 2.2.b_c $2$ 2.4.d_i 2.2.d_g $2$ 2.4.d_i
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.2.ad_g $2$ 2.4.d_i 2.2.b_c $2$ 2.4.d_i 2.2.d_g $2$ 2.4.d_i 2.2.ad_g $4$ 2.16.h_y 2.2.ab_e $8$ 2.256.ab_asm 2.2.b_e $8$ 2.256.ab_asm