# Properties

 Label 3.3.ae_e_a 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: $( 1 - 3 x + 3 x^{2} )( 1 - x - 2 x^{2} - 3 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0734519173280$, $\pm0.166666666667$, $\pm0.740118583995$ 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:

• $x^4+2x^3z+x^2y^2+x^2z^2+xy^2z+2y^4+y^3z+z^4=0$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 4 336 11200 646464 14147284 406425600 11275287244 281074789632 7732512961600 206706589004496

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 2 12 98 240 764 2352 6530 19956 59282

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 2.3.ab_ac 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.ak 2 $\times$ 1.729.cc. The endomorphism algebra for each factor is: 1.729.ak 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$ 1.729.cc : 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 $\times$ 2.9.af_q. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.ai 2 $\times$ 1.27.a. The endomorphism algebra for each factor is: 1.27.ai 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$ 1.27.a : $$\Q(\sqrt{-3})$$.

## 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_ac_m $2$ 3.9.ai_bo_afi 3.3.c_ac_am $2$ 3.9.ai_bo_afi 3.3.e_e_a $2$ 3.9.ai_bo_afi 3.3.ab_b_ag $3$ (not in LMFDB) 3.3.ab_e_aj $3$ (not in LMFDB) 3.3.c_ac_am $3$ (not in LMFDB) 3.3.c_k_m $3$ (not in LMFDB) 3.3.f_q_bh $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.ac_ac_m $2$ 3.9.ai_bo_afi 3.3.c_ac_am $2$ 3.9.ai_bo_afi 3.3.e_e_a $2$ 3.9.ai_bo_afi 3.3.ab_b_ag $3$ (not in LMFDB) 3.3.ab_e_aj $3$ (not in LMFDB) 3.3.c_ac_am $3$ (not in LMFDB) 3.3.c_k_m $3$ (not in LMFDB) 3.3.f_q_bh $3$ (not in LMFDB) 3.3.af_q_abh $6$ (not in LMFDB) 3.3.ad_i_ap $6$ (not in LMFDB) 3.3.ac_k_am $6$ (not in LMFDB) 3.3.a_i_a $6$ (not in LMFDB) 3.3.b_b_g $6$ (not in LMFDB) 3.3.b_e_j $6$ (not in LMFDB) 3.3.d_i_p $6$ (not in LMFDB) 3.3.ad_ac_p $12$ (not in LMFDB) 3.3.a_ac_a $12$ (not in LMFDB) 3.3.d_ac_ap $12$ (not in LMFDB)