# Properties

 Label 3.3.ai_be_aco Base Field $\F_{3}$ Dimension $3$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $3$ L-polynomial: $( 1 - 2 x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.304086723985$ Angle rank: $1$ (numerical) Jacobians: 0

This isogeny class is not simple.

## Newton polygon $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1/2, 1/2, 1]$

## 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 $A(\F_{q^r})$ 2 588 29792 794976 17772722 420424704 10821854822 288077043072 7700377178144 205877327739468

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 6 38 114 296 792 2264 6690 19874 59046

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 1.3.ac 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.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.3.ac : $$\Q(\sqrt{-2})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.cc 2 . The endomorphism algebra for each factor is: 1.729.abu : $$\Q(\sqrt{-2})$$. 1.729.cc 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
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.ad 2 $\times$ 1.9.c. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.9.c : $$\Q(\sqrt{-2})$$.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 1.27.k. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 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.ae_g_ag $2$ 3.9.ae_y_acc 3.3.ac_a_g $2$ 3.9.ae_y_acc 3.3.c_a_ag $2$ 3.9.ae_y_acc 3.3.e_g_g $2$ 3.9.ae_y_acc 3.3.i_be_co $2$ 3.9.ae_y_acc 3.3.af_p_abe $3$ (not in LMFDB) 3.3.ac_a_g $3$ (not in LMFDB) 3.3.ac_j_am $3$ (not in LMFDB) 3.3.b_d_g $3$ (not in LMFDB) 3.3.e_g_g $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.ae_g_ag $2$ 3.9.ae_y_acc 3.3.ac_a_g $2$ 3.9.ae_y_acc 3.3.c_a_ag $2$ 3.9.ae_y_acc 3.3.e_g_g $2$ 3.9.ae_y_acc 3.3.i_be_co $2$ 3.9.ae_y_acc 3.3.af_p_abe $3$ (not in LMFDB) 3.3.ac_a_g $3$ (not in LMFDB) 3.3.ac_j_am $3$ (not in LMFDB) 3.3.b_d_g $3$ (not in LMFDB) 3.3.e_g_g $3$ (not in LMFDB) 3.3.ac_g_ag $4$ (not in LMFDB) 3.3.c_g_g $4$ (not in LMFDB) 3.3.ab_d_ag $6$ (not in LMFDB) 3.3.c_j_m $6$ (not in LMFDB) 3.3.f_p_be $6$ (not in LMFDB) 3.3.ac_ad_m $12$ (not in LMFDB) 3.3.c_ad_am $12$ (not in LMFDB) 3.3.ac_d_a $24$ (not in LMFDB) 3.3.c_d_a $24$ (not in LMFDB)