# Properties

 Label 2.9.ah_be Base field $\F_{3^{2}}$ Dimension $2$ $p$-rank $1$ Ordinary No Supersingular No Simple No Geometrically simple No Primitive Yes Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3^{2}}$ Dimension: $2$ L-polynomial: $( 1 - 4 x + 9 x^{2} )( 1 - 3 x + 9 x^{2} )$ Frobenius angles: $\pm0.267720472801$, $\pm0.333333333333$ Angle rank: $1$ (numerical) Jacobians: 0

This isogeny class is not simple.

## Newton polygon

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

## Point counts

This isogeny class is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 42 7644 606816 44640960 3486431802 281402424576 22847046425322 1852792867511040 150105608160199776 12158088943167680604

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 3 93 828 6801 59043 529506 4776747 43041441 387448812 3486905853

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3^{2}}$
 The isogeny class factors as 1.9.ae $\times$ 1.9.ad 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 $\overline{\F}_{3^{2}}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.bs $\times$ 1.729.cc. The endomorphism algebra for each factor is: 1.729.bs : $$\Q(\sqrt{-5})$$. 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{6}}$.

## Base change

This is a primitive isogeny class.

## Twists

Below are some of the twists of this isogeny class.
 Twist Extension Degree Common base change 2.9.ab_g $2$ 2.81.l_gy 2.9.b_g $2$ 2.81.l_gy 2.9.h_be $2$ 2.81.l_gy 2.9.c_ag $3$ 2.729.du_frm
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 2.9.ab_g $2$ 2.81.l_gy 2.9.b_g $2$ 2.81.l_gy 2.9.h_be $2$ 2.81.l_gy 2.9.c_ag $3$ 2.729.du_frm 2.9.ak_bq $6$ (not in LMFDB) 2.9.ac_ag $6$ (not in LMFDB) 2.9.k_bq $6$ (not in LMFDB) 2.9.ae_s $12$ (not in LMFDB) 2.9.e_s $12$ (not in LMFDB)