Properties

Label 2.2.b_ab
Base field $\F_{2}$
Dimension $2$
$p$-rank $2$
Ordinary Yes
Supersingular No
Simple Yes
Geometrically simple No
Primitive Yes
Principally polarizable No
Contains a Jacobian No

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Invariants

Base field:  $\F_{2}$
Dimension:  $2$
L-polynomial:  $1 + x - x^{2} + 2 x^{3} + 4 x^{4}$
Frobenius angles:  $\pm0.281693394748$, $\pm0.948360061415$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{-7})\)
Galois group:  $C_2^2$
Jacobians:  0

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 is not principally polarizable, and therefore does not contain a Jacobian.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 7 7 196 259 1477 3136 14749 58275 268324 1109227

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 4 2 19 18 44 47 116 226 523 1082

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-7})\).
Endomorphism algebra over $\overline{\F}_{2}$
The base change of $A$ to $\F_{2^{3}}$ is 1.8.f 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{3}}$.

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_ab$2$2.4.ad_f
2.2.ac_f$3$2.8.k_bp
2.2.a_d$6$2.64.as_ib
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.2.ab_ab$2$2.4.ad_f
2.2.ac_f$3$2.8.k_bp
2.2.a_d$6$2.64.as_ib
2.2.c_f$6$2.64.as_ib
2.2.a_ad$12$(not in LMFDB)