Properties

Label 2.3.ac_d
Base field $\F_{3}$
Dimension $2$
$p$-rank $1$
Ordinary no
Supersingular no
Simple no
Geometrically simple no
Primitive yes
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{3}$
Dimension:  $2$
L-polynomial:  $( 1 - 3 x + 3 x^{2} )( 1 + x + 3 x^{2} )$
  $1 - 2x + 3x^{2} - 6x^{3} + 9x^{4}$
Frobenius angles:  $\pm0.166666666667$, $\pm0.593214749339$
Angle rank:  $1$ (numerical)
Jacobians:  2

This isogeny class is not simple.

Newton polygon

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

Point counts

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $5$ $105$ $560$ $6825$ $74525$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $2$ $12$ $20$ $84$ $302$ $774$ $2186$ $6756$ $19820$ $58332$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ad $\times$ 1.3.b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{3}$
The base change of $A$ to $\F_{3^{6}}$ is 1.729.ak $\times$ 1.729.cc. The endomorphism algebra for each factor is:
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Remainder of endomorphism lattice by field

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.ae_j$2$2.9.c_d
2.3.c_d$2$2.9.c_d
2.3.e_j$2$2.9.c_d
2.3.b_g$3$2.27.ai_cc
2.3.e_j$3$2.27.ai_cc
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.ae_j$2$2.9.c_d
2.3.c_d$2$2.9.c_d
2.3.e_j$2$2.9.c_d
2.3.b_g$3$2.27.ai_cc
2.3.e_j$3$2.27.ai_cc
2.3.ab_g$6$2.729.bs_bji