# Properties

 Label 4.3.ah_t_ay_y 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: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - x - 2 x^{2} - 3 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0734519173280$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.740118583995$ Angle rank: $1$ (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 $A(\F_{q^r})$ 4 2352 313600 58828224 3833913964 318637670400 25583626756636 1867179827525376 152206785136134400 12155794379587396272

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 -1 12 107 267 818 2433 6611 19956 59039

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 2.3.ab_ac 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.ab_ac : $$\Q(\sqrt{-3}, \sqrt{-11})$$.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.ak 2 $\times$ 1.729.cc 2 . The endomorphism algebra for each factor is: 1.729.ak 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$ 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 $\times$ 2.9.af_q. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.9.af_q : $$\Q(\sqrt{-3}, \sqrt{-11})$$.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.ai 2 $\times$ 1.27.a 2 . The endomorphism algebra for each factor is: 1.27.ai 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-11})$$$)$ 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$

## 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.af_h_m_abw $2$ (not in LMFDB) 4.3.ab_af_a_y $2$ (not in LMFDB) 4.3.b_af_a_y $2$ (not in LMFDB) 4.3.f_h_am_abw $2$ (not in LMFDB) 4.3.h_t_y_y $2$ (not in LMFDB) 4.3.ae_h_am_y $3$ (not in LMFDB) 4.3.ae_k_ay_bz $3$ (not in LMFDB) 4.3.ab_af_a_y $3$ (not in LMFDB) 4.3.ab_e_aj_g $3$ (not in LMFDB) 4.3.ab_h_am_y $3$ (not in LMFDB) 4.3.c_b_ag_am $3$ (not in LMFDB) 4.3.c_e_a_ad $3$ (not in LMFDB) 4.3.c_n_s_ci $3$ (not in LMFDB) 4.3.f_h_am_abw $3$ (not in LMFDB) 4.3.f_t_bw_ds $3$ (not in LMFDB) 4.3.i_bi_ds_hn $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.af_h_m_abw $2$ (not in LMFDB) 4.3.ab_af_a_y $2$ (not in LMFDB) 4.3.b_af_a_y $2$ (not in LMFDB) 4.3.f_h_am_abw $2$ (not in LMFDB) 4.3.h_t_y_y $2$ (not in LMFDB) 4.3.ae_h_am_y $3$ (not in LMFDB) 4.3.ae_k_ay_bz $3$ (not in LMFDB) 4.3.ab_af_a_y $3$ (not in LMFDB) 4.3.ab_e_aj_g $3$ (not in LMFDB) 4.3.ab_h_am_y $3$ (not in LMFDB) 4.3.c_b_ag_am $3$ (not in LMFDB) 4.3.c_e_a_ad $3$ (not in LMFDB) 4.3.c_n_s_ci $3$ (not in LMFDB) 4.3.f_h_am_abw $3$ (not in LMFDB) 4.3.f_t_bw_ds $3$ (not in LMFDB) 4.3.i_bi_ds_hn $3$ (not in LMFDB) 4.3.ab_b_ag_m $4$ (not in LMFDB) 4.3.b_b_g_m $4$ (not in LMFDB) 4.3.ai_bi_ads_hn $6$ (not in LMFDB) 4.3.ag_u_abw_dp $6$ (not in LMFDB) 4.3.af_t_abw_ds $6$ (not in LMFDB) 4.3.ad_l_ay_bw $6$ (not in LMFDB) 4.3.ac_b_g_am $6$ (not in LMFDB) 4.3.ac_e_a_ad $6$ (not in LMFDB) 4.3.ac_n_as_ci $6$ (not in LMFDB) 4.3.a_c_a_d $6$ (not in LMFDB) 4.3.a_l_a_bw $6$ (not in LMFDB) 4.3.b_e_j_g $6$ (not in LMFDB) 4.3.b_h_m_y $6$ (not in LMFDB) 4.3.d_l_y_bw $6$ (not in LMFDB) 4.3.e_h_m_y $6$ (not in LMFDB) 4.3.e_k_y_bz $6$ (not in LMFDB) 4.3.g_u_bw_dp $6$ (not in LMFDB) 4.3.ag_k_m_acf $12$ (not in LMFDB) 4.3.ad_b_g_am $12$ (not in LMFDB) 4.3.ac_b_g_ay $12$ (not in LMFDB) 4.3.ac_k_am_bn $12$ (not in LMFDB) 4.3.ab_ai_d_be $12$ (not in LMFDB) 4.3.a_al_a_bw $12$ (not in LMFDB) 4.3.a_ai_a_bh $12$ (not in LMFDB) 4.3.a_ac_a_d $12$ (not in LMFDB) 4.3.a_ab_a_am $12$ (not in LMFDB) 4.3.a_b_a_am $12$ (not in LMFDB) 4.3.a_i_a_bh $12$ (not in LMFDB) 4.3.b_ai_ad_be $12$ (not in LMFDB) 4.3.c_b_ag_ay $12$ (not in LMFDB) 4.3.c_k_m_bn $12$ (not in LMFDB) 4.3.d_b_ag_am $12$ (not in LMFDB) 4.3.g_k_am_acf $12$ (not in LMFDB) 4.3.ac_h_ag_s $24$ (not in LMFDB) 4.3.ab_ac_ad_s $24$ (not in LMFDB) 4.3.a_af_a_s $24$ (not in LMFDB) 4.3.a_f_a_s $24$ (not in LMFDB) 4.3.b_ac_d_s $24$ (not in LMFDB) 4.3.c_h_g_s $24$ (not in LMFDB)