# Properties

 Label 4.3.ah_v_abk_cc Base Field $\F_{3}$ Dimension $4$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $4$ L-polynomial: $( 1 + 2 x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{3}$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.695913276015$ Angle rank: $1$ (numerical)

This isogeny class is not simple.

## Newton polygon

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

## 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 $A(\F_{q^r})$ 6 4116 395136 72342816 4896017706 329612967936 26564029141866 1913695797127296 148676055648494976 12107028012374894676

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 3 18 123 327 846 2517 6771 19494 58803

## Decomposition and endomorphism algebra

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

## 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.al_cf_agy_oo $2$ (not in LMFDB) 4.3.af_j_a_as $2$ (not in LMFDB) 4.3.ab_ad_a_s $2$ (not in LMFDB) 4.3.b_ad_a_s $2$ (not in LMFDB) 4.3.f_j_a_as $2$ (not in LMFDB) 4.3.h_v_bk_cc $2$ (not in LMFDB) 4.3.l_cf_gy_oo $2$ (not in LMFDB) 4.3.ae_j_as_bk $3$ (not in LMFDB) 4.3.ab_ad_a_s $3$ (not in LMFDB) 4.3.ab_g_aj_s $3$ (not in LMFDB) 4.3.c_d_a_a $3$ (not in LMFDB) 4.3.c_m_s_cc $3$ (not in LMFDB) 4.3.f_j_a_as $3$ (not in LMFDB) 4.3.f_s_bt_dm $3$ (not in LMFDB) 4.3.i_bh_dm_gy $3$ (not in LMFDB) 4.3.l_cf_gy_oo $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.al_cf_agy_oo $2$ (not in LMFDB) 4.3.af_j_a_as $2$ (not in LMFDB) 4.3.ab_ad_a_s $2$ (not in LMFDB) 4.3.b_ad_a_s $2$ (not in LMFDB) 4.3.f_j_a_as $2$ (not in LMFDB) 4.3.h_v_bk_cc $2$ (not in LMFDB) 4.3.l_cf_gy_oo $2$ (not in LMFDB) 4.3.ae_j_as_bk $3$ (not in LMFDB) 4.3.ab_ad_a_s $3$ (not in LMFDB) 4.3.ab_g_aj_s $3$ (not in LMFDB) 4.3.c_d_a_a $3$ (not in LMFDB) 4.3.c_m_s_cc $3$ (not in LMFDB) 4.3.f_j_a_as $3$ (not in LMFDB) 4.3.f_s_bt_dm $3$ (not in LMFDB) 4.3.i_bh_dm_gy $3$ (not in LMFDB) 4.3.l_cf_gy_oo $3$ (not in LMFDB) 4.3.af_p_abe_cc $4$ (not in LMFDB) 4.3.ab_d_ag_s $4$ (not in LMFDB) 4.3.b_d_g_s $4$ (not in LMFDB) 4.3.f_p_be_cc $4$ (not in LMFDB) 4.3.ai_bh_adm_gy $6$ (not in LMFDB) 4.3.af_s_abt_dm $6$ (not in LMFDB) 4.3.ac_d_a_a $6$ (not in LMFDB) 4.3.ac_m_as_cc $6$ (not in LMFDB) 4.3.b_g_j_s $6$ (not in LMFDB) 4.3.e_j_s_bk $6$ (not in LMFDB) 4.3.c_d_aj_as $9$ (not in LMFDB) 4.3.c_d_j_s $9$ (not in LMFDB) 4.3.af_g_p_acc $12$ (not in LMFDB) 4.3.ac_a_g_as $12$ (not in LMFDB) 4.3.ac_j_am_bk $12$ (not in LMFDB) 4.3.ab_ag_d_s $12$ (not in LMFDB) 4.3.b_ag_ad_s $12$ (not in LMFDB) 4.3.c_a_ag_as $12$ (not in LMFDB) 4.3.c_j_m_bk $12$ (not in LMFDB) 4.3.f_g_ap_acc $12$ (not in LMFDB) 4.3.ac_d_aj_s $18$ (not in LMFDB) 4.3.ac_d_j_as $18$ (not in LMFDB) 4.3.af_m_ap_s $24$ (not in LMFDB) 4.3.ac_g_ag_s $24$ (not in LMFDB) 4.3.ab_a_ad_s $24$ (not in LMFDB) 4.3.b_a_d_s $24$ (not in LMFDB) 4.3.c_g_g_s $24$ (not in LMFDB) 4.3.f_m_p_s $24$ (not in LMFDB)