Properties

Label 1.4.e
Base Field $\F_{2^2}$
Dimension $1$
$p$-rank $0$
Principally polarizable
Contains a Jacobian

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Invariants

Base field:  $\F_{2^2}$
Dimension:  $1$
Weil polynomial:  $( 1 + 2 x )^{2}$
Frobenius angles:  $1.0$, $1.0$
Angle rank:  $0$ (numerical)
Number field:  \(\Q\)
Galois group:  Trivial

This isogeny class is simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

Point counts

This isogeny class contains a Jacobian, and hence is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 9 9 81 225 1089 3969 16641 65025 263169 1046529

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 9 9 81 225 1089 3969 16641 65025 263169 1046529

Decomposition

This is a simple isogeny class.

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^2}$.

SubfieldPrimitive Model
$\F_{2}$1.2.a