# Properties

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

## Invariants

 Base field: $\F_{3}$ Dimension: $3$ L-polynomial: $( 1 - x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.406785250661$ Angle rank: $1$ (numerical) Jacobians: 0

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 is principally polarizable, but does not contain a Jacobian.

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 3 735 28224 621075 15642933 442552320 11691927831 294564072075 7574065265088 202567769810175

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 9 36 93 267 828 2433 6837 19548 58089

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 1.3.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: 1.3.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.3.ab : $$\Q(\sqrt{-11})$$.
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 2 $\times$ 1.9.f. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 1.9.f : $$\Q(\sqrt{-11})$$.
• 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.af_m_av $2$ 3.9.ab_g_bb 3.3.ab_a_d $2$ 3.9.ab_g_bb 3.3.b_a_ad $2$ 3.9.ab_g_bb 3.3.f_m_v $2$ 3.9.ab_g_bb 3.3.h_y_bz $2$ 3.9.ab_g_bb 3.3.ae_m_ay $3$ (not in LMFDB) 3.3.ab_j_ag $3$ (not in LMFDB) 3.3.c_g_m $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.af_m_av $2$ 3.9.ab_g_bb 3.3.ab_a_d $2$ 3.9.ab_g_bb 3.3.b_a_ad $2$ 3.9.ab_g_bb 3.3.f_m_v $2$ 3.9.ab_g_bb 3.3.h_y_bz $2$ 3.9.ab_g_bb 3.3.ae_m_ay $3$ (not in LMFDB) 3.3.ab_j_ag $3$ (not in LMFDB) 3.3.c_g_m $3$ (not in LMFDB) 3.3.ab_g_ad $4$ (not in LMFDB) 3.3.b_g_d $4$ (not in LMFDB) 3.3.af_m_av $6$ (not in LMFDB) 3.3.ac_g_am $6$ (not in LMFDB) 3.3.b_j_g $6$ (not in LMFDB) 3.3.e_m_y $6$ (not in LMFDB) 3.3.ab_ad_g $12$ (not in LMFDB) 3.3.b_ad_ag $12$ (not in LMFDB) 3.3.ab_d_a $24$ (not in LMFDB) 3.3.b_d_a $24$ (not in LMFDB)