# Properties

 Label 3.3.af_l_as Base field $\F_{3}$ Dimension $3$ $p$-rank $2$ Ordinary No Supersingular No Simple No Geometrically simple No Primitive Yes Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $3$ L-polynomial: $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 2 x^{2} - 6 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.116139763599$, $\pm0.166666666667$, $\pm0.616139763599$ 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 $A(\F_{q^r})$ 4 560 12208 582400 18710924 417025280 10982404724 299987251200 7718757861136 205043660654000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -1 7 14 87 309 784 2295 6959 19922 58807

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 2.3.ac_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 $\F_{3^{12}}$ is 1.531441.acec $\times$ 1.531441.sk 2 . The endomorphism algebra for each factor is: 1.531441.acec : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 1.531441.sk 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-5})$$$)$
All geometric endomorphisms are defined over $\F_{3^{12}}$.
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.a_ac. 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.ao_du. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{3^{4}}$  The base change of $A$ to $\F_{3^{4}}$ is 1.81.ac 2 $\times$ 1.81.j. The endomorphism algebra for each factor is: 1.81.ac 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-5})$$$)$ 1.81.j : $$\Q(\sqrt{-3})$$.
• Endomorphism algebra over $\F_{3^{6}}$  The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc $\times$ 2.729.a_sk. The endomorphism algebra for each factor is: 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 2.729.a_sk : $$\Q(i, \sqrt{5})$$.

## 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_ab_g $2$ 3.9.ad_h_g 3.3.f_l_s $2$ 3.9.ad_h_g 3.3.ac_f_am $3$ (not in LMFDB) 3.3.b_ab_ag $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.ab_ab_g $2$ 3.9.ad_h_g 3.3.f_l_s $2$ 3.9.ad_h_g 3.3.ac_f_am $3$ (not in LMFDB) 3.3.b_ab_ag $3$ (not in LMFDB) 3.3.ac_f_am $6$ (not in LMFDB) 3.3.c_f_m $6$ (not in LMFDB) 3.3.ad_ab_m $8$ (not in LMFDB) 3.3.ad_h_am $8$ (not in LMFDB) 3.3.d_ab_am $8$ (not in LMFDB) 3.3.d_h_m $8$ (not in LMFDB) 3.3.a_ab_a $24$ (not in LMFDB) 3.3.a_h_a $24$ (not in LMFDB)