# Properties

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

## Invariants

 Base field: $\F_{3}$ Dimension: $3$ L-polynomial: $( 1 - 2 x + 3 x^{2} )( 1 - 2 x + x^{2} - 6 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0292466093486$, $\pm0.304086723985$, $\pm0.637420057318$ Angle rank: $1$ (numerical) Jacobians: 0

This isogeny class is not simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $3$ Slopes: $[0, 0, 0, 1, 1, 1]$

## 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 $A(\F_{q^r})$ 6 684 12312 530784 14114166 320013504 9669223122 282430166400 7552438699464 205891158689964

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 10 18 82 240 592 2016 6562 19494 59050

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ac $\times$ 2.3.ac_b 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.abu 3 and its endomorphism algebra is $\mathrm{M}_{3}($$$\Q(\sqrt{-2})$$$)$
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.c $\times$ 2.9.ac_af. 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.ak 2 $\times$ 1.27.k. The endomorphism algebra for each factor is: 1.27.ak 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.27.k : $$\Q(\sqrt{-2})$$.

## 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.a_a_ak $2$ 3.9.a_a_abu 3.3.a_a_k $2$ 3.9.a_a_abu 3.3.e_i_o $2$ 3.9.a_a_abu 3.3.c_f_e $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.a_a_ak $2$ 3.9.a_a_abu 3.3.a_a_k $2$ 3.9.a_a_abu 3.3.e_i_o $2$ 3.9.a_a_abu 3.3.c_f_e $3$ (not in LMFDB) 3.3.ag_v_abs $6$ (not in LMFDB) 3.3.ac_f_ae $6$ (not in LMFDB) 3.3.g_v_bs $6$ (not in LMFDB) 3.3.ac_b_e $12$ (not in LMFDB) 3.3.c_b_ae $12$ (not in LMFDB) 3.3.ag_t_abo $24$ (not in LMFDB) 3.3.ac_d_ai $24$ (not in LMFDB) 3.3.c_d_i $24$ (not in LMFDB) 3.3.g_t_bo $24$ (not in LMFDB)