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:  $( 1 - x + 2 x^{2} )( 1 + 2 x^{2} )$
  $1 - x + 4 x^{2} - 2 x^{3} + 4 x^{4}$
Frobenius angles:  $\pm0.384973271919$, $\pm0.5$
Angle rank:  $1$ (numerical)
Jacobians:  $0$
Isomorphism classes:  1

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $6$ $72$ $126$ $144$ $726$

Point counts of the (virtual) curve

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

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{2}}$.

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:

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