Properties

Label 2.2.d_g
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:  $( 1 + x + 2 x^{2} )( 1 + 2 x + 2 x^{2} )$
Frobenius angles:  $\pm0.615026728081$, $\pm0.750000000000$
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
$A(\F_{q^r})$ 20 40 20 400 1100 3640 16820 64800 276860 992200

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 6 8 0 24 36 56 132 256 540 968

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.b $\times$ 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:
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Remainder of endomorphism lattice by field

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.b_c$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.b_c$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