Properties

Label 2.9.ai_bi
Base field $\F_{3^{2}}$
Dimension $2$
$p$-rank $2$
Ordinary yes
Supersingular no
Simple no
Geometrically simple no
Primitive no
Principally polarizable yes
Contains a Jacobian yes

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Invariants

Base field:  $\F_{3^{2}}$
Dimension:  $2$
L-polynomial:  $( 1 - 4 x + 9 x^{2} )^{2}$
  $1 - 8 x + 34 x^{2} - 72 x^{3} + 81 x^{4}$
Frobenius angles:  $\pm0.267720472801$, $\pm0.267720472801$
Angle rank:  $1$ (numerical)
Jacobians:  $3$

This isogeny class is not simple, not 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

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $36$ $7056$ $599076$ $45158400$ $3514829796$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $2$ $86$ $818$ $6878$ $59522$ $530486$ $4774898$ $43023038$ $387398402$ $3486909206$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{2}}$.

Endomorphism algebra over $\F_{3^{2}}$
The isogeny class factors as 1.9.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{2}}$.

SubfieldPrimitive Model
$\F_{3}$2.3.a_ae

Twists

Below are some of the twists of this isogeny class.

TwistExtension degreeCommon base change
2.9.a_c$2$2.81.e_gk
2.9.i_bi$2$2.81.e_gk
2.9.e_h$3$2.729.dk_fao

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

TwistExtension degreeCommon base change
2.9.a_c$2$2.81.e_gk
2.9.i_bi$2$2.81.e_gk
2.9.e_h$3$2.729.dk_fao
2.9.a_ac$4$(not in LMFDB)
2.9.ae_h$6$(not in LMFDB)