# Properties

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

## Invariants

 Base field: $\F_{3}$ Dimension: $3$ L-polynomial: $( 1 + 3 x^{2} )( 1 - 2 x + 3 x^{2} )^{2}$ Frobenius angles: $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.5$ Angle rank: $1$ (numerical) Jacobians: 1

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 contains the Jacobians of 1 curves, and hence is principally polarizable:

• $y^2=2x^8+x^7+x^5+x^4+2x^3+2x+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 16 2304 40432 589824 14289616 366799104 9667467952 272734617600 7774705143184 210999098052864

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 20 48 92 240 692 2016 6332 20064 60500

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ac 2 $\times$ 1.3.a 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.a : $$\Q(\sqrt{-3})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{2}}$ is 1.9.c 2 $\times$ 1.9.g. The endomorphism algebra for each factor is: 1.9.c 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 1.9.g : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
All geometric endomorphisms are defined over $\F_{3^{2}}$.

## 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_f_a $2$ 3.9.k_cd_hw 3.3.e_n_y $2$ 3.9.k_cd_hw 3.3.ah_z_acc $3$ (not in LMFDB) 3.3.ab_ac_j $3$ (not in LMFDB) 3.3.ab_b_g $3$ (not in LMFDB) 3.3.c_e_m $3$ (not in LMFDB) 3.3.f_k_p $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.a_f_a $2$ 3.9.k_cd_hw 3.3.e_n_y $2$ 3.9.k_cd_hw 3.3.ah_z_acc $3$ (not in LMFDB) 3.3.ab_ac_j $3$ (not in LMFDB) 3.3.ab_b_g $3$ (not in LMFDB) 3.3.c_e_m $3$ (not in LMFDB) 3.3.f_k_p $3$ (not in LMFDB) 3.3.a_b_a $4$ (not in LMFDB) 3.3.af_k_ap $6$ (not in LMFDB) 3.3.ad_f_ag $6$ (not in LMFDB) 3.3.ac_e_am $6$ (not in LMFDB) 3.3.ab_b_g $6$ (not in LMFDB) 3.3.b_ac_aj $6$ (not in LMFDB) 3.3.b_b_ag $6$ (not in LMFDB) 3.3.d_f_g $6$ (not in LMFDB) 3.3.h_z_cc $6$ (not in LMFDB) 3.3.ae_l_ay $8$ (not in LMFDB) 3.3.e_l_y $8$ (not in LMFDB) 3.3.ad_b_g $12$ (not in LMFDB) 3.3.d_b_ag $12$ (not in LMFDB) 3.3.ah_x_abw $24$ (not in LMFDB) 3.3.ab_ab_a $24$ (not in LMFDB) 3.3.b_ab_a $24$ (not in LMFDB) 3.3.h_x_bw $24$ (not in LMFDB)