Properties

Label 2.3.ac_h
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:  $( 1 - x + 3 x^{2} )^{2}$
  $1 - 2x + 7x^{2} - 6x^{3} + 9x^{4}$
Frobenius angles:  $\pm0.406785250661$, $\pm0.406785250661$
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 Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Point counts of the abelian variety

$r$ $1$ $2$ $3$ $4$ $5$
$A(\F_{q^r})$ $9$ $225$ $1296$ $5625$ $45369$

Point counts of the curve

$r$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$
$C(\F_{q^r})$ $2$ $20$ $44$ $68$ $182$ $710$ $2354$ $6788$ $19412$ $58100$

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
The isogeny class factors as 1.3.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
All geometric endomorphisms are defined over $\F_{3}$.

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.a_f$2$2.9.k_br
2.3.c_h$2$2.9.k_br
2.3.b_ac$3$2.27.q_eo
Below is a list of all twists of this isogeny class.
TwistExtension degreeCommon base change
2.3.a_f$2$2.9.k_br
2.3.c_h$2$2.9.k_br
2.3.b_ac$3$2.27.q_eo
2.3.a_af$4$2.81.ao_id
2.3.ab_ac$6$2.729.au_chy