# Properties

 Label 3.3.af_r_abi 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: $( 1 - x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )^{2}$ Frobenius angles: $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.406785250661$ Angle rank: $2$ (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 $A(\F_{q^r})$ 12 2160 51984 691200 12474132 336856320 10034195484 284453683200 7720988424048 207593257330800

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 19 56 103 209 628 2099 6607 19928 59539

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ac 2 $\times$ 1.3.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.3.ac 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.3.ab : $$\Q(\sqrt{-11})$$.
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.ad_j_ao $2$ 3.9.j_bz_ha 3.3.ab_f_ac $2$ 3.9.j_bz_ha 3.3.b_f_c $2$ 3.9.j_bz_ha 3.3.d_j_o $2$ 3.9.j_bz_ha 3.3.f_r_bi $2$ 3.9.j_bz_ha 3.3.b_c_l $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.ad_j_ao $2$ 3.9.j_bz_ha 3.3.ab_f_ac $2$ 3.9.j_bz_ha 3.3.b_f_c $2$ 3.9.j_bz_ha 3.3.d_j_o $2$ 3.9.j_bz_ha 3.3.f_r_bi $2$ 3.9.j_bz_ha 3.3.b_c_l $3$ (not in LMFDB) 3.3.ab_b_c $4$ (not in LMFDB) 3.3.b_b_ac $4$ (not in LMFDB) 3.3.ad_g_an $6$ (not in LMFDB) 3.3.ab_c_al $6$ (not in LMFDB) 3.3.d_g_n $6$ (not in LMFDB) 3.3.af_p_abg $8$ (not in LMFDB) 3.3.ad_h_aq $8$ (not in LMFDB) 3.3.d_h_q $8$ (not in LMFDB) 3.3.f_p_bg $8$ (not in LMFDB)