Properties

Label 2.2.a_ab
Base Field $\F_{2}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2}$
Dimension:  $2$
L-polynomial:  $1 - x^{2} + 4 x^{4}$
Frobenius angles:  $\pm0.209784688372$, $\pm0.790215311628$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{5})\)
Galois group:  $C_2^2$
Jacobians:  1

This isogeny class is simple but not geometrically simple.

Newton polygon

This isogeny class is ordinary.

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

Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 4 16 76 576 964 5776 16636 57600 261364 929296

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 3 3 9 31 33 87 129 223 513 903

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{5})\).
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$
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.ad_f$3$2.8.a_l
2.2.d_f$3$2.8.a_l
2.2.a_b$4$2.16.o_dd
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.2.ad_f$3$2.8.a_l
2.2.d_f$3$2.8.a_l
2.2.a_b$4$2.16.o_dd
2.2.ad_f$6$2.64.w_jp