Properties

Label 2.2.ab_e
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 No

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Invariants

Base field:  $\F_{2}$
Dimension:  $2$
L-polynomial:  $( 1 - x + 2 x^{2} )( 1 + 2 x^{2} )$
Frobenius angles:  $\pm0.384973271919$, $\pm0.5$
Angle rank:  $1$ (numerical)
Jacobians:  0

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 6 72 126 144 726 4536 18318 64800 265734 1054152

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 12 14 8 22 72 142 256 518 1032

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.ab $\times$ 1.2.a 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 $\F_{2^{2}}$ is 1.4.d $\times$ 1.4.e. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{2^{2}}$.

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.b_e$2$2.4.h_u
2.2.ad_g$8$2.256.ab_asm
2.2.ab_c$8$2.256.ab_asm
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.2.b_e$2$2.4.h_u
2.2.ad_g$8$2.256.ab_asm
2.2.ab_c$8$2.256.ab_asm
2.2.b_c$8$2.256.ab_asm
2.2.d_g$8$2.256.ab_asm