# Properties

 Label 3.3.ae_i_am 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: $\pm0.166666666667$, $\pm0.235082516458$, $\pm0.648854628963$ Angle rank: $2$ (numerical) Jacobians: 2

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 2 curves, and hence is principally polarizable:

• $y^2=2x^8+2x^7+2x^6+x^5+x^3+2x^2+2x+2$
• $x^4+2x^3z+x^2z^2+xy^2z+y^4+z^4=0$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 8 896 17024 792064 19535848 396591104 10717358296 285171554304 7431198793088 204231505874816

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 10 24 114 320 748 2240 6626 19176 58570

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 2.3.ab_c 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.cc $\times$ 2.729.abk_bas. The endomorphism algebra for each factor is: 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 2.729.abk_bas : 4.0.3757.1.
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.d_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.a $\times$ 2.27.ae_ak. The endomorphism algebra for each factor is:

## 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_c_a $2$ 3.9.a_q_g 3.3.c_c_a $2$ 3.9.a_q_g 3.3.e_i_m $2$ 3.9.a_q_g 3.3.ab_f_ag $3$ (not in LMFDB) 3.3.c_c_a $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.ac_c_a $2$ 3.9.a_q_g 3.3.c_c_a $2$ 3.9.a_q_g 3.3.e_i_m $2$ 3.9.a_q_g 3.3.ab_f_ag $3$ (not in LMFDB) 3.3.c_c_a $3$ (not in LMFDB) 3.3.ac_c_a $6$ (not in LMFDB) 3.3.ab_f_ag $6$ (not in LMFDB) 3.3.b_f_g $6$ (not in LMFDB)