Properties

Label 1.2.ac
Base Field $\F_{2}$
Dimension $1$
Ordinary No
$p$-rank $0$
Principally polarizable Yes
Contains a Jacobian Yes

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Invariants

Base field:  $\F_{2}$
Dimension:  $1$
Weil polynomial:  $1 - 2 x + 2 x^{2}$
Frobenius angles:  $\pm0.25$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-1}) \)
Galois group:  $C_2$

This isogeny class is simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

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
$A(\F_{q^r})$ 1 5 13 25 41 65 113 225 481 1025

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 1 5 13 25 41 65 113 225 481 1025

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.