Properties

Label 3.3.af_k_ap
Base Field $\F_{3}$
Dimension $3$
Ordinary No
$p$-rank $2$
Contains a Jacobian No

Learn more about

Invariants

Base field:  $\F_{3}$
Dimension:  $3$
Weil polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + x^{2} - 6 x^{3} + 9 x^{4} )$
Frobenius angles:  $\pm0.0292466093486$, $\pm0.166666666667$, $\pm0.637420057318$
Angle rank:  $1$ (numerical)

Newton polygon

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

Point counts

This isogeny class does not contain a Jacobian, and it is unknown whether it is principally polarizable.

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 3 399 9072 503139 15805533 366799104 10437424959 287405575275 7480235652624 203383749396639

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ -1 5 8 77 269 692 2183 6677 19304 58325

Decomposition

1.3.ad $\times$ 2.3.ac_b

Base change

This is a primitive isogeny class.