# Properties

 Label 3.3.ae_m_ay Base Field $\F_{3}$ Dimension $3$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{3}$ Dimension: $3$ L-polynomial: $( 1 - 3 x + 3 x^{2} )( 1 - x + 3 x^{2} )( 1 + 3 x^{2} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.406785250661$, $\pm0.5$ Angle rank: $1$ (numerical) Jacobians: 1

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $1$ Slopes: $[0, 1/2, 1/2, 1/2, 1/2, 1]$

## Point counts

This isogeny class contains the Jacobians of 1 curves, and hence is principally polarizable:

• $y^2=2x^8+2x^7+2x^6+2x^2+x+2$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 12 1680 28224 436800 14084412 442552320 11274543012 283788960000 7574065265088 205078897808400

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 18 36 66 240 828 2352 6594 19548 58818

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 1.3.ab $\times$ 1.3.a 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.ak $\times$ 1.729.cc 2 . The endomorphism algebra for each factor is: 1.729.ak : $$\Q(\sqrt{-11})$$. 1.729.cc 2 : $\mathrm{M}_{2}(B)$, where $B$ is 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.f $\times$ 1.9.g. The endomorphism algebra for each factor is: 1.9.ad : $$\Q(\sqrt{-3})$$. 1.9.f : $$\Q(\sqrt{-11})$$. 1.9.g : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 1.27.i. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.27.i : $$\Q(\sqrt{-11})$$.

## 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_g_am $2$ 3.9.i_y_cc 3.3.c_g_m $2$ 3.9.i_y_cc 3.3.e_m_y $2$ 3.9.i_y_cc 3.3.ah_y_abz $3$ (not in LMFDB) 3.3.ab_a_d $3$ (not in LMFDB) 3.3.ab_j_ag $3$ (not in LMFDB) 3.3.c_g_m $3$ (not in LMFDB) 3.3.f_m_v $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.ac_g_am $2$ 3.9.i_y_cc 3.3.c_g_m $2$ 3.9.i_y_cc 3.3.e_m_y $2$ 3.9.i_y_cc 3.3.ah_y_abz $3$ (not in LMFDB) 3.3.ab_a_d $3$ (not in LMFDB) 3.3.ab_j_ag $3$ (not in LMFDB) 3.3.c_g_m $3$ (not in LMFDB) 3.3.f_m_v $3$ (not in LMFDB) 3.3.af_m_av $6$ (not in LMFDB) 3.3.b_a_ad $6$ (not in LMFDB) 3.3.b_j_g $6$ (not in LMFDB) 3.3.h_y_bz $6$ (not in LMFDB) 3.3.ab_ad_g $12$ (not in LMFDB) 3.3.ab_g_ad $12$ (not in LMFDB) 3.3.b_ad_ag $12$ (not in LMFDB) 3.3.b_g_d $12$ (not in LMFDB) 3.3.ab_d_a $24$ (not in LMFDB) 3.3.b_d_a $24$ (not in LMFDB)