Properties

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

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Invariants

Base field:  $\F_{2^2}$
Dimension:  $2$
Weil polynomial:  $1 - 3 x + 5 x^{2} - 12 x^{3} + 16 x^{4}$
Frobenius angles:  $\pm0.103279877171$, $\pm0.563386789496$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-3}, \sqrt{-7})\)
Galois group:  $V_4$

This isogeny class is simple.

Newton polygon

This isogeny class is ordinary.

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

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
$A(\F_{q^r})$ 7 259 3136 58275 1109227 17172736 267001147 4324529475 69244764736 1100771362579

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 2 18 47 226 1082 4191 16298 65986 264143 1049778

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}$2.2.ab_ab
$\F_{2}$2.2.b_ab