# Properties

 Label 3.3.ag_v_abs 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: $( 1 - 2 x + 3 x^{2} )^{3}$ Frobenius angles: $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.304086723985$ 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 $A(\F_{q^r})$ 8 1728 54872 884736 14172488 320013504 9287485208 278189309952 7849750559624 210984921816768

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 16 58 124 238 592 1930 6460 20254 60496

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ac 3 and its endomorphism algebra is $\mathrm{M}_{3}($$$\Q(\sqrt{-2})$$$)$
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.
 Twist Extension Degree Common base change 3.3.ac_f_ae $2$ 3.9.g_bn_em 3.3.c_f_e $2$ 3.9.g_bn_em 3.3.g_v_bs $2$ 3.9.g_bn_em 3.3.a_a_k $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.ac_f_ae $2$ 3.9.g_bn_em 3.3.c_f_e $2$ 3.9.g_bn_em 3.3.g_v_bs $2$ 3.9.g_bn_em 3.3.a_a_k $3$ (not in LMFDB) 3.3.ac_b_e $4$ (not in LMFDB) 3.3.c_b_ae $4$ (not in LMFDB) 3.3.ae_i_ao $6$ (not in LMFDB) 3.3.a_a_ak $6$ (not in LMFDB) 3.3.e_i_o $6$ (not in LMFDB) 3.3.ag_t_abo $8$ (not in LMFDB) 3.3.ac_d_ai $8$ (not in LMFDB) 3.3.c_d_i $8$ (not in LMFDB) 3.3.g_t_bo $8$ (not in LMFDB)