Properties

Label 2.2.ac_f
Base field $\F_{2}$
Dimension $2$
$p$-rank $2$
Ordinary yes
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:  $( 1 - x + 2 x^{2} )^{2}$
  $1 - 2 x + 5 x^{2} - 4 x^{3} + 4 x^{4}$
Frobenius angles:  $\pm0.384973271919$, $\pm0.384973271919$
Angle rank:  $1$ (numerical)
Jacobians:  0

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

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

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $4$ $64$ $196$ $256$ $484$

Point counts of the (virtual) curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $11$ $19$ $15$ $11$ $47$ $155$ $319$ $523$ $911$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The isogeny class factors as 1.2.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
All geometric endomorphisms are defined over $\F_{2}$.

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.a_d$2$2.4.g_r
2.2.c_f$2$2.4.g_r
2.2.b_ab$3$2.8.k_bp
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.2.a_d$2$2.4.g_r
2.2.c_f$2$2.4.g_r
2.2.b_ab$3$2.8.k_bp
2.2.a_ad$4$2.16.ac_bh
2.2.ab_ab$6$2.64.as_ib