Properties

Label 2.2.a_b
Base field $\F_{2}$
Dimension $2$
$p$-rank $2$
Ordinary Yes
Supersingular No
Simple Yes
Geometrically simple No
Primitive Yes
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:  $\pm0.290215311628$, $\pm0.709784688372$
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
$A(\F_{q^r})$ 6 36 54 576 1086 2916 16134 57600 262926 1179396

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 3 7 9 31 33 43 129 223 513 1147

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.b 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 is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.2.a_ab$4$2.16.o_dd
2.2.ad_f$12$(not in LMFDB)
2.2.d_f$12$(not in LMFDB)