Base field $\F_{3}$
Dimension $1$
$p$-rank $0$
Ordinary no
Supersingular yes
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes


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Base field:  $\F_{3}$
Dimension:  $1$
L-polynomial:  $1 - 3 x + 3 x^{2}$
Frobenius angles:  $\pm0.166666666667$
Angle rank:  $0$ (numerical)
Number field:  \(\Q(\sqrt{-3}) \)
Galois group:  $C_2$
Jacobians:  1

This isogeny class is simple and geometrically simple.

Newton polygon

This isogeny class is supersingular.

$p$-rank:  $0$
Slopes:  $[1/2, 1/2]$

Point counts

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $1$ $7$ $28$ $91$ $271$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $1$ $7$ $28$ $91$ $271$ $784$ $2269$ $6643$ $19684$ $58807$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \).
Endomorphism algebra over $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{6}}$ is the simple isogeny class and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.


Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change

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.