# Properties

 Label 3.3.ag_t_abo 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: $( 1 - 2 x + 3 x^{2} )( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0540867239847$, $\pm0.304086723985$, $\pm0.445913276015$ 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 $A(\F_{q^r})$ 4 816 23788 443904 11926244 363956400 10295801996 278189309952 7586211740356 207582049122096

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 12 34 68 198 684 2154 6460 19582 59532

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ac $\times$ 2.3.ae_i 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 $\F_{3^{8}}$ is 1.6561.abi 3 and its endomorphism algebra is $\mathrm{M}_{3}($$$\Q(\sqrt{-2})$$$)$
All geometric endomorphisms are defined over $\F_{3^{8}}$.
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.a_ao. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{3^{4}}$  The base change of $A$ to $\F_{3^{4}}$ is 1.81.ao 2 $\times$ 1.81.o. The endomorphism algebra for each factor is: 1.81.ao 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.81.o : $$\Q(\sqrt{-2})$$.

## 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.ac_d_ai $2$ 3.9.c_af_abc 3.3.c_d_i $2$ 3.9.c_af_abc 3.3.g_t_bo $2$ 3.9.c_af_abc
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.ac_d_ai $2$ 3.9.c_af_abc 3.3.c_d_i $2$ 3.9.c_af_abc 3.3.g_t_bo $2$ 3.9.c_af_abc 3.3.ag_v_abs $8$ (not in LMFDB) 3.3.ac_b_e $8$ (not in LMFDB) 3.3.ac_f_ae $8$ (not in LMFDB) 3.3.c_b_ae $8$ (not in LMFDB) 3.3.c_f_e $8$ (not in LMFDB) 3.3.g_v_bs $8$ (not in LMFDB) 3.3.ae_i_ao $24$ (not in LMFDB) 3.3.a_a_ak $24$ (not in LMFDB) 3.3.a_a_k $24$ (not in LMFDB) 3.3.e_i_o $24$ (not in LMFDB)