Properties

Label 3.2.ad_c_b
Base Field $\F_{2}$
Dimension $3$
Ordinary No
$p$-rank $3$

Learn more about

Invariants

Base field:  $\F_{2}$
Dimension:  $3$
Weil polynomial:  $1 - 3 x + 2 x^{2} + x^{3} + 4 x^{4} - 12 x^{5} + 8 x^{6}$
Frobenius angles:  $\pm0.0992589862044$, $\pm0.18645529951$, $\pm0.757883870938$
Angle rank:  $1$ (numerical)
Number field:  \(\Q(\zeta_{7})\)
Galois group:  $C_6$

This isogeny class is simple.

Newton polygon

This isogeny class is ordinary.

$p$-rank:  $3$
Slopes:  $[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 1 29 301 8149 34861 375347 2863288 16436533 149808001 1062528419

Point counts of the (virtual) curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 0 3 28 35 87 168 252 570 1015

Decomposition

This is a simple isogeny class.

Base change

This is a primitive isogeny class.

Additional information

This is the isogeny class of the Jacobian of a function field of class number 1. It also appears as a sporadic example in the classification of abelian varieties with one rational point.