Properties

Label 2.3.ac_f
Base field $\F_{3}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple yes
Geometrically simple yes
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
L-polynomial:  $1 - 2 x + 5 x^{2} - 6 x^{3} + 9 x^{4}$
Frobenius angles:  $\pm0.254551732336$, $\pm0.538152604671$
Angle rank:  $2$ (numerical)
Number field:  4.0.4672.2
Galois group:  $D_{4}$
Jacobians:  $1$
Isomorphism classes:  1

This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $7$ $161$ $868$ $6601$ $69167$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $2$ $16$ $32$ $84$ $282$ $766$ $2046$ $6308$ $19760$ $59296$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3}$.

Endomorphism algebra over $\F_{3}$
The endomorphism algebra of this simple isogeny class is 4.0.4672.2.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

TwistExtension degreeCommon base change
2.3.c_f$2$2.9.g_t