Properties

Label 2.9.ak_br
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:  $( 1 - 5 x + 9 x^{2} )^{2}$
Frobenius angles:  $\pm0.186429498677$, $\pm0.186429498677$
Angle rank:  $1$ (numerical)
Jacobians:  1

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 1 curves, and hence is principally polarizable:

Point counts of the abelian variety

$r$ 1 2 3 4 5 6 7 8 9 10
$A(\F_{q^r})$ 25 5625 547600 44555625 3543225625 283875840000 22900866685225 1853050666055625 150078465576144400 12156915630659765625

Point counts of the curve

$r$ 1 2 3 4 5 6 7 8 9 10
$C(\F_{q^r})$ 0 68 750 6788 60000 534158 4788000 43047428 387378750 3486569348

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{2}}$
The isogeny class factors as 1.9.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
All geometric endomorphisms are defined over $\F_{3^{2}}$.

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_af

Twists

Below are some of the twists of this isogeny class.
TwistExtension DegreeCommon base change
2.9.a_ah$2$2.81.ao_id
2.9.k_br$2$2.81.ao_id
2.9.f_q$3$2.729.u_chy
Below is a list of all twists of this isogeny class.
TwistExtension DegreeCommon base change
2.9.a_ah$2$2.81.ao_id
2.9.k_br$2$2.81.ao_id
2.9.f_q$3$2.729.u_chy
2.9.a_h$4$(not in LMFDB)
2.9.af_q$6$(not in LMFDB)