# Properties

 Label 5.3.ak_by_agk_pv_abes Base Field $\F_{3}$ Dimension $5$ Ordinary No $p$-rank $3$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $5$ L-polynomial: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 11 x^{2} - 22 x^{3} + 33 x^{4} - 36 x^{5} + 27 x^{6} )$ Frobenius angles: $\pm0.132091637252$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.376445424065$, $\pm0.544359499442$ Angle rank: $3$ (numerical)

This isogeny class is not simple.

## Newton polygon

 $p$-rank: $3$ Slopes: $[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$

## 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 $A(\F_{q^r})$ 10 65660 17271520 3772823600 1079586372050 252212275911680 55134360098537590 12944967590790726400 3045381532607929772320 711872319466612931318300

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 10 30 86 304 880 2402 6974 20280 58550

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 3.3.ae_l_aw 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}($$$\Q(\sqrt{-3})$$$)$ 3.3.ae_l_aw : 6.0.5169344.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 $\times$ 3.729.bq_bex_fga. 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 $\infty$. 3.729.bq_bex_fga : 6.0.5169344.1.
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 $\times$ 3.9.g_l_i. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 3.9.g_l_i : 6.0.5169344.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 3.27.c_x_ji. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 3.27.c_x_ji : 6.0.5169344.1.

## 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 5.3.ae_i_ak_j_ag $2$ (not in LMFDB) 5.3.ac_c_ac_d_g $2$ (not in LMFDB) 5.3.c_c_c_d_ag $2$ (not in LMFDB) 5.3.e_i_k_j_g $2$ (not in LMFDB) 5.3.k_by_gk_pv_bes $2$ (not in LMFDB) 5.3.ah_bd_adk_ic_apm $3$ (not in LMFDB) 5.3.ae_i_ak_j_ag $3$ (not in LMFDB) 5.3.ae_r_abu_ee_ahw $3$ (not in LMFDB) 5.3.ab_f_ae_g_ag $3$ (not in LMFDB) 5.3.c_c_c_d_ag $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ae_i_ak_j_ag $2$ (not in LMFDB) 5.3.ac_c_ac_d_g $2$ (not in LMFDB) 5.3.c_c_c_d_ag $2$ (not in LMFDB) 5.3.e_i_k_j_g $2$ (not in LMFDB) 5.3.k_by_gk_pv_bes $2$ (not in LMFDB) 5.3.ah_bd_adk_ic_apm $3$ (not in LMFDB) 5.3.ae_i_ak_j_ag $3$ (not in LMFDB) 5.3.ae_r_abu_ee_ahw $3$ (not in LMFDB) 5.3.ab_f_ae_g_ag $3$ (not in LMFDB) 5.3.c_c_c_d_ag $3$ (not in LMFDB) 5.3.ae_o_abi_cx_afi $4$ (not in LMFDB) 5.3.e_o_bi_cx_fi $4$ (not in LMFDB) 5.3.b_f_e_g_g $6$ (not in LMFDB) 5.3.e_r_bu_ee_hw $6$ (not in LMFDB) 5.3.h_bd_dk_ic_pm $6$ (not in LMFDB) 5.3.ae_f_c_ay_ci $12$ (not in LMFDB) 5.3.e_f_ac_ay_aci $12$ (not in LMFDB) 5.3.ae_l_aw_bq_acu $24$ (not in LMFDB) 5.3.e_l_w_bq_cu $24$ (not in LMFDB)