Properties

Label 2.4.a_d
Base Field $\F_{2^{2}}$
Dimension $2$
Ordinary Yes
$p$-rank $2$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2^{2}}$
Dimension:  $2$
L-polynomial:  $1 + 3 x^{2} + 16 x^{4}$
Frobenius angles:  $\pm0.311178646770$, $\pm0.688821353230$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{5}, \sqrt{-11})\)
Galois group:  $C_2^2$
Jacobians:  6

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 6 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 20 400 3980 78400 1050500 15840400 268421180 4292870400 68719312820 1103550250000

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 5 23 65 303 1025 3863 16385 65503 262145 1052423

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{2^{2}}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{5}, \sqrt{-11})\).
Endomorphism algebra over $\overline{\F}_{2^{2}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.d 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-55}) \)$)$
All geometric endomorphisms are defined over $\F_{2^{4}}$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.4.a_ad$4$2.256.bu_bob