# Properties

 Label 2.2.b_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: $( 1 - x + 2 x^{2} )( 1 + 2 x + 2 x^{2} )$ $1 + x + 2x^{2} + 2x^{3} + 4x^{4}$ Frobenius angles: $\pm0.384973271919$, $\pm0.750000000000$ 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 Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2+xy=x^5+x^3+x^2+x$

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $10$ $40$ $70$ $400$ $550$

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $8$ $10$ $24$ $14$ $56$ $158$ $256$ $550$ $968$

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
 The isogeny class factors as 1.2.ab $\times$ 1.2.c 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.
TwistExtension degreeCommon base change
2.2.ad_g$2$2.4.d_i
2.2.ab_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.
TwistExtension degreeCommon base change
2.2.ad_g$2$2.4.d_i
2.2.ab_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