# Properties

 Label 3.3.ae_l_ay 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: $( 1 + 3 x^{2} )( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0540867239847$, $\pm0.445913276015$, $\pm0.5$ 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:

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

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 8 1088 17528 295936 12024808 417166400 10717038424 272734617600 7513685815496 207595996716608

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 16 24 36 200 784 2240 6332 19392 59536

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.a $\times$ 2.3.ae_i 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 $\F_{3^{4}}$ is 1.81.as $\times$ 1.81.ao 2 . The endomorphism algebra for each factor is: 1.81.as : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 1.81.ao 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-2})$$$)$
All geometric endomorphisms are defined over $\F_{3^{4}}$.
Remainder of endomorphism lattice by field
• Endomorphism algebra over $\F_{3^{2}}$  The base change of $A$ to $\F_{3^{2}}$ is 1.9.g $\times$ 2.9.a_ao. The endomorphism algebra for each factor is: 1.9.g : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 2.9.a_ao : $$\Q(\zeta_{8})$$.

## 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.e_l_y $2$ 3.9.g_af_adg 3.3.ah_x_abw $3$ (not in LMFDB) 3.3.ab_ab_a $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.e_l_y $2$ 3.9.g_af_adg 3.3.ah_x_abw $3$ (not in LMFDB) 3.3.ab_ab_a $3$ (not in LMFDB) 3.3.b_ab_a $6$ (not in LMFDB) 3.3.h_x_bw $6$ (not in LMFDB) 3.3.ae_n_ay $8$ (not in LMFDB) 3.3.a_b_a $8$ (not in LMFDB) 3.3.a_f_a $8$ (not in LMFDB) 3.3.e_n_y $8$ (not in LMFDB) 3.3.ah_z_acc $24$ (not in LMFDB) 3.3.af_k_ap $24$ (not in LMFDB) 3.3.ad_b_g $24$ (not in LMFDB) 3.3.ad_f_ag $24$ (not in LMFDB) 3.3.ac_e_am $24$ (not in LMFDB) 3.3.ab_ac_j $24$ (not in LMFDB) 3.3.ab_b_g $24$ (not in LMFDB) 3.3.b_ac_aj $24$ (not in LMFDB) 3.3.b_b_ag $24$ (not in LMFDB) 3.3.c_e_m $24$ (not in LMFDB) 3.3.d_b_ag $24$ (not in LMFDB) 3.3.d_f_g $24$ (not in LMFDB) 3.3.f_k_p $24$ (not in LMFDB) 3.3.h_z_cc $24$ (not in LMFDB)