Properties

Label 2.3.a_c
Base field $\F_{3}$
Dimension $2$
$p$-rank $2$
Ordinary yes
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:  $( 1 - 2 x + 3 x^{2} )( 1 + 2 x + 3 x^{2} )$
  $1 + 2 x^{2} + 9 x^{4}$
Frobenius angles:  $\pm0.304086723985$, $\pm0.695913276015$
Angle rank:  $1$ (numerical)
Jacobians:  2

This isogeny class is not simple.

Newton polygon

This isogeny class is ordinary.

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

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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $12$ $144$ $684$ $9216$ $59532$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $4$ $14$ $28$ $110$ $244$ $638$ $2188$ $6494$ $19684$ $60014$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ac $\times$ 1.3.c 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 $\F_{3^{2}}$ is 1.9.c 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
All geometric endomorphisms are defined over $\F_{3^{2}}$.

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_k$2$2.9.e_w
2.3.e_k$2$2.9.e_w
2.3.a_ac$4$2.81.bc_nu
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.ae_k$2$2.9.e_w
2.3.e_k$2$2.9.e_w
2.3.a_ac$4$2.81.bc_nu
2.3.ac_b$6$2.729.ado_fhm
2.3.c_b$6$2.729.ado_fhm
2.3.ae_i$8$(not in LMFDB)
2.3.e_i$8$(not in LMFDB)