# Properties

 Label 3.3.af_n_ay Base Field $\F_{3}$ Dimension $3$ Ordinary No $p$-rank $2$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $3$ L-polynomial: $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 4 x^{2} - 6 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.210767374595$, $\pm0.567777800232$ Angle rank: $2$ (numerical) Jacobians: 0

This isogeny class is not simple.

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

## 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 $A(\F_{q^r})$ 6 924 19656 624624 20196546 447999552 10434001038 283069602816 7624602557208 202574870579004

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 11 26 95 329 836 2183 6575 19682 58091

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 2.3.ac_e 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 $\F_{3^{6}}$ is 1.729.cc $\times$ 2.729.ca_czy. The endomorphism algebra for each factor is: 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 2.729.ca_czy : 4.0.7488.1.
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad $\times$ 2.9.e_k. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 2.27.ac_bc. The endomorphism algebra for each factor is:

## 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 3.3.ab_b_a $2$ 3.9.b_h_bq 3.3.f_n_y $2$ 3.9.b_h_bq 3.3.ac_h_am $3$ (not in LMFDB) 3.3.b_b_a $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.ab_b_a $2$ 3.9.b_h_bq 3.3.f_n_y $2$ 3.9.b_h_bq 3.3.ac_h_am $3$ (not in LMFDB) 3.3.b_b_a $3$ (not in LMFDB) 3.3.ac_h_am $6$ (not in LMFDB) 3.3.c_h_m $6$ (not in LMFDB)