# Properties

 Label 4.3.ai_bg_adg_gj Base Field $\F_{3}$ Dimension $4$ Ordinary No $p$-rank $2$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $4$ L-polynomial: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 2 x + 5 x^{2} - 6 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.254551732336$, $\pm0.538152604671$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

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

## 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 $A(\F_{q^r})$ 7 7889 680512 54662881 5079693647 343587786752 23073342565007 1827315445641777 150688359407419456 12108904870630602209

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 10 32 102 336 874 2208 6470 19760 58810

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 2.3.ac_f 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}($$$\Q(\sqrt{-3})$$$)$ 2.3.ac_f : 4.0.4672.2.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 $\times$ 2.729.bk_bww. The endomorphism algebra for each factor is: 1.729.cc 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 2.729.bk_bww : 4.0.4672.2.
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 $\times$ 2.9.g_t. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.9.g_t : 4.0.4672.2.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 2.27.e_ba. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.27.e_ba : 4.0.4672.2.

## 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 4.3.ae_i_am_v $2$ (not in LMFDB) 4.3.ac_c_a_d $2$ (not in LMFDB) 4.3.c_c_a_d $2$ (not in LMFDB) 4.3.e_i_m_v $2$ (not in LMFDB) 4.3.i_bg_dg_gj $2$ (not in LMFDB) 4.3.af_r_abq_dg $3$ (not in LMFDB) 4.3.ac_c_a_d $3$ (not in LMFDB) 4.3.ac_l_as_bw $3$ (not in LMFDB) 4.3.b_f_g_m $3$ (not in LMFDB) 4.3.e_i_m_v $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ae_i_am_v $2$ (not in LMFDB) 4.3.ac_c_a_d $2$ (not in LMFDB) 4.3.c_c_a_d $2$ (not in LMFDB) 4.3.e_i_m_v $2$ (not in LMFDB) 4.3.i_bg_dg_gj $2$ (not in LMFDB) 4.3.af_r_abq_dg $3$ (not in LMFDB) 4.3.ac_c_a_d $3$ (not in LMFDB) 4.3.ac_l_as_bw $3$ (not in LMFDB) 4.3.b_f_g_m $3$ (not in LMFDB) 4.3.e_i_m_v $3$ (not in LMFDB) 4.3.ac_i_am_bh $4$ (not in LMFDB) 4.3.c_i_m_bh $4$ (not in LMFDB) 4.3.ac_c_a_d $6$ (not in LMFDB) 4.3.ab_f_ag_m $6$ (not in LMFDB) 4.3.c_l_s_bw $6$ (not in LMFDB) 4.3.f_r_bq_dg $6$ (not in LMFDB) 4.3.ac_ab_g_am $12$ (not in LMFDB) 4.3.c_ab_ag_am $12$ (not in LMFDB) 4.3.ac_f_ag_s $24$ (not in LMFDB) 4.3.c_f_g_s $24$ (not in LMFDB)