Properties

Label 2.27.au_fy
Base Field $\F_{3^3}$
Dimension $2$
$p$-rank $2$
Principally polarizable
Contains a Jacobian

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Invariants

Base field:  $\F_{3^3}$
Dimension:  $2$
Weil polynomial:  $( 1 - 10 x + 27 x^{2} )^{2}$
Frobenius angles:  $\pm0.0877398280459$, $\pm0.0877398280459$
Angle rank:  $1$ (numerical)

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
$A(\F_{q^r})$ 324 467856 380016036 281731654656 205849551061764 150097166713147536 109420491473516900196 79766798892731688960000 58149803503012262724708804 42391169226695938352674321296

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 8 638 19304 530126 14346008 387427022 10460496824 282430796318 7625606205608 205891185285278

Decomposition

1.27.ak 2

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}$2.3.ac_b
$\F_{3}$2.3.e_k