# Properties

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

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## Invariants

 Base field: $\F_{3}$ Dimension: $4$ L-polynomial: $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )( 1 - 2 x + 2 x^{2} - 6 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.116139763599$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.616139763599$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

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

## 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 $A(\F_{q^r})$ 8 6720 463904 55910400 4528043608 285245291520 23085014729848 1958316775833600 153402593732216864 12206659206053928000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 9 24 101 307 738 2209 6925 20112 59289

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 1.3.ac $\times$ 2.3.ac_c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{12}}$ is 1.531441.acec $\times$ 1.531441.azi $\times$ 1.531441.sk 2 . The endomorphism algebra for each factor is: 1.531441.acec : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 1.531441.azi : $$\Q(\sqrt{-2})$$. 1.531441.sk 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-5})$$$)$
All geometric endomorphisms are defined over $\F_{3^{12}}$.
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 $\times$ 1.9.c $\times$ 2.9.a_ac. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 1.27.k $\times$ 2.27.ao_du. The endomorphism algebra for each factor is:
• Endomorphism algebra over $\F_{3^{4}}$  The base change of $A$ to $\F_{3^{4}}$ is 1.81.ac 2 $\times$ 1.81.j $\times$ 1.81.o. The endomorphism algebra for each factor is: 1.81.ac 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-5})$$$)$ 1.81.j : $$\Q(\sqrt{-3})$$. 1.81.o : $$\Q(\sqrt{-2})$$.
• Endomorphism algebra over $\F_{3^{6}}$  The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.cc $\times$ 2.729.a_sk. The endomorphism algebra for each factor is: 1.729.abu : $$\Q(\sqrt{-2})$$. 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 2.729.a_sk : $$\Q(i, \sqrt{5})$$.

## 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.ad_e_al_be $2$ (not in LMFDB) 4.3.ad_e_f_as $2$ (not in LMFDB) 4.3.ab_a_ab_g $2$ (not in LMFDB) 4.3.b_a_b_g $2$ (not in LMFDB) 4.3.d_e_af_as $2$ (not in LMFDB) 4.3.d_e_l_be $2$ (not in LMFDB) 4.3.h_y_cd_dy $2$ (not in LMFDB) 4.3.ae_m_abc_cc $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ad_e_al_be $2$ (not in LMFDB) 4.3.ad_e_f_as $2$ (not in LMFDB) 4.3.ab_a_ab_g $2$ (not in LMFDB) 4.3.b_a_b_g $2$ (not in LMFDB) 4.3.d_e_af_as $2$ (not in LMFDB) 4.3.d_e_l_be $2$ (not in LMFDB) 4.3.h_y_cd_dy $2$ (not in LMFDB) 4.3.ae_m_abc_cc $3$ (not in LMFDB) 4.3.a_e_ai_g $6$ (not in LMFDB) 4.3.a_e_i_g $6$ (not in LMFDB) 4.3.e_m_bc_cc $6$ (not in LMFDB) 4.3.af_i_f_abe $8$ (not in LMFDB) 4.3.af_q_abj_co $8$ (not in LMFDB) 4.3.ab_ae_b_s $8$ (not in LMFDB) 4.3.ab_e_ah_s $8$ (not in LMFDB) 4.3.b_ae_ab_s $8$ (not in LMFDB) 4.3.b_e_h_s $8$ (not in LMFDB) 4.3.f_i_af_abe $8$ (not in LMFDB) 4.3.f_q_bj_co $8$ (not in LMFDB) 4.3.ac_c_c_ag $24$ (not in LMFDB) 4.3.ac_k_ao_bq $24$ (not in LMFDB) 4.3.c_c_ac_ag $24$ (not in LMFDB) 4.3.c_k_o_bq $24$ (not in LMFDB)