Properties

Label 1.27.ai
Base Field $\F_{3^3}$
Dimension $1$
$p$-rank $1$
Principally polarizable
Contains a Jacobian

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Invariants

Base field:  $\F_{3^3}$
Dimension:  $1$
Weil polynomial:  $1 - 8 x + 27 x^{2}$
Frobenius angles:  $\pm0.220355751984$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\sqrt{-11}) \)
Galois group:  $C_2$

This isogeny class is simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $1$
Slopes:  $[0, 1]$

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
$A(\F_{q^r})$ 20 720 19820 532800 14356100 387441360 10460325980 282428755200 7625591969780 205891109067600

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 20 720 19820 532800 14356100 387441360 10460325980 282428755200 7625591969780 205891109067600

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 $\F_{3^3}$.

SubfieldPrimitive Model
$\F_{3}$1.3.b