Properties

Label 2.2.ad_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:  $( 1 - 2 x + 2 x^{2} )( 1 - x + 2 x^{2} )$
  $1 - 3x + 6x^{2} - 6x^{3} + 4x^{4}$
Frobenius angles:  $\pm0.250000000000$, $\pm0.384973271919$
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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $2$ $40$ $182$ $400$ $902$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $0$ $8$ $18$ $24$ $30$ $56$ $126$ $256$ $486$ $968$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.ac $\times$ 1.2.ab 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.ab_c$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.
TwistExtension degreeCommon base change
2.2.ab_c$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.ab_c$4$2.16.h_y
2.2.b_c$4$2.16.h_y
2.2.d_g$4$2.16.h_y
2.2.ab_e$8$2.256.ab_asm
2.2.b_e$8$2.256.ab_asm