# Properties

 Label 3.3.ae_k_as 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: $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )( 1 + x + 3 x^{2} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.593214749339$ Angle rank: $2$ (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:

• $y^2=2x^8+x^7+x^6+2x^4+x^2+2x+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 10 1260 21280 655200 18035050 386104320 10039666990 289464739200 7753580353120 205065119328300

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 14 30 98 300 728 2100 6722 20010 58814

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 1.3.ac $\times$ 1.3.b 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.ak $\times$ 1.729.cc. The endomorphism algebra for each factor is: 1.729.abu : $$\Q(\sqrt{-2})$$. 1.729.ak : $$\Q(\sqrt{-11})$$. 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 $\times$ 1.9.c $\times$ 1.9.f. 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 $\times$ 1.27.a $\times$ 1.27.k. 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.ag_u_abq $2$ 3.9.e_q_bq 3.3.ac_e_ag $2$ 3.9.e_q_bq 3.3.a_c_ag $2$ 3.9.e_q_bq 3.3.a_c_g $2$ 3.9.e_q_bq 3.3.c_e_g $2$ 3.9.e_q_bq 3.3.e_k_s $2$ 3.9.e_q_bq 3.3.g_u_bq $2$ 3.9.e_q_bq 3.3.ab_h_ag $3$ (not in LMFDB) 3.3.c_e_g $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.ag_u_abq $2$ 3.9.e_q_bq 3.3.ac_e_ag $2$ 3.9.e_q_bq 3.3.a_c_ag $2$ 3.9.e_q_bq 3.3.a_c_g $2$ 3.9.e_q_bq 3.3.c_e_g $2$ 3.9.e_q_bq 3.3.e_k_s $2$ 3.9.e_q_bq 3.3.g_u_bq $2$ 3.9.e_q_bq 3.3.ab_h_ag $3$ (not in LMFDB) 3.3.c_e_g $3$ (not in LMFDB) 3.3.ad_l_as $6$ (not in LMFDB) 3.3.b_h_g $6$ (not in LMFDB) 3.3.d_l_s $6$ (not in LMFDB)