Properties

Label 2.27.an_dl
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:  $1 - 13 x + 89 x^{2} - 351 x^{3} + 729 x^{4}$
Frobenius angles:  $\pm0.154461042763$, $\pm0.380600815323$
Angle rank:  $2$ (numerical)
Number field:  4.0.10933.1
Galois group:  $D_4$

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
$A(\F_{q^r})$ 455 538265 391801865 282656408125 205869481308400 150099896680966505 109422115050212922685 79766961890306814703125 58149761029767385837890395 42391152604967519660811795200

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 15 739 19905 531867 14347400 387434071 10460652035 282431373443 7625600635785 205891104554614

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}$2.3.ab_ab