# Properties

 Label 3.3.af_k_ap Base Field $\F_{3}$ Dimension $3$ Ordinary No $p$-rank $2$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $3$ L-polynomial: $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + x^{2} - 6 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0292466093486$, $\pm0.166666666667$, $\pm0.637420057318$ Angle rank: $1$ (numerical) Jacobians: 0

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 is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 3 399 9072 503139 15805533 366799104 10437424959 287405575275 7480235652624 203383749396639

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 5 8 77 269 692 2183 6677 19304 58325

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 2.3.ac_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 2 $\times$ 1.729.cc. The endomorphism algebra for each factor is: 1.729.abu 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 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.ac_af. 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.ak 2 $\times$ 1.27.a. The endomorphism algebra for each factor is: 1.27.ak 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$ 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.ab_ac_j $2$ 3.9.af_k_av 3.3.b_ac_aj $2$ 3.9.af_k_av 3.3.f_k_p $2$ 3.9.af_k_av 3.3.ac_e_am $3$ (not in LMFDB) 3.3.b_ac_aj $3$ (not in LMFDB) 3.3.b_b_ag $3$ (not in LMFDB) 3.3.e_n_y $3$ (not in LMFDB) 3.3.h_z_cc $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.ab_ac_j $2$ 3.9.af_k_av 3.3.b_ac_aj $2$ 3.9.af_k_av 3.3.f_k_p $2$ 3.9.af_k_av 3.3.ac_e_am $3$ (not in LMFDB) 3.3.b_ac_aj $3$ (not in LMFDB) 3.3.b_b_ag $3$ (not in LMFDB) 3.3.e_n_y $3$ (not in LMFDB) 3.3.h_z_cc $3$ (not in LMFDB) 3.3.ah_z_acc $6$ (not in LMFDB) 3.3.ae_n_ay $6$ (not in LMFDB) 3.3.ad_f_ag $6$ (not in LMFDB) 3.3.ab_b_g $6$ (not in LMFDB) 3.3.a_f_a $6$ (not in LMFDB) 3.3.c_e_m $6$ (not in LMFDB) 3.3.d_f_g $6$ (not in LMFDB) 3.3.ad_b_g $12$ (not in LMFDB) 3.3.a_b_a $12$ (not in LMFDB) 3.3.d_b_ag $12$ (not in LMFDB) 3.3.ah_x_abw $24$ (not in LMFDB) 3.3.ae_l_ay $24$ (not in LMFDB) 3.3.ab_ab_a $24$ (not in LMFDB) 3.3.b_ab_a $24$ (not in LMFDB) 3.3.e_l_y $24$ (not in LMFDB) 3.3.h_x_bw $24$ (not in LMFDB)