# Properties

 Label 3.3.af_p_abe 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: $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )( 1 + 3 x^{2} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.5$ 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 $A(\F_{q^r})$ 8 1344 29792 559104 16002008 420424704 10435530344 277539225600 7700377178144 208429482617664

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 15 38 87 269 792 2183 6447 19874 59775

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 1.3.ac $\times$ 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:
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 $\times$ 1.9.c $\times$ 1.9.g. The endomorphism algebra for each factor is: 1.9.ad : $$\Q(\sqrt{-3})$$. 1.9.c : $$\Q(\sqrt{-2})$$. 1.9.g : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
• 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.ab_d_ag $2$ 3.9.f_p_cc 3.3.b_d_g $2$ 3.9.f_p_cc 3.3.f_p_be $2$ 3.9.f_p_cc 3.3.ai_be_aco $3$ (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.ab_d_ag $2$ 3.9.f_p_cc 3.3.b_d_g $2$ 3.9.f_p_cc 3.3.f_p_be $2$ 3.9.f_p_cc 3.3.ai_be_aco $3$ (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.ae_g_ag $6$ (not in LMFDB) 3.3.c_a_ag $6$ (not in LMFDB) 3.3.c_j_m $6$ (not in LMFDB) 3.3.i_be_co $6$ (not in LMFDB) 3.3.ac_ad_m $12$ (not in LMFDB) 3.3.ac_g_ag $12$ (not in LMFDB) 3.3.c_ad_am $12$ (not in LMFDB) 3.3.c_g_g $12$ (not in LMFDB) 3.3.ac_d_a $24$ (not in LMFDB) 3.3.c_d_a $24$ (not in LMFDB)