# Properties

 Label 4.3.ah_u_abe_bn 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: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - x - x^{2} - 3 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.126866938441$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.718153680921$ 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 $A(\F_{q^r})$ 5 3185 356720 67283125 4435836400 330847811840 26727999291355 1913590813168125 151807494193431440 12208565650804140800

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 1 15 117 302 847 2531 6773 19905 59296

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 2.3.ab_ab 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_ab : 4.0.10933.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 $\times$ 2.729.j_jt. 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.j_jt : 4.0.10933.1.
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 $\times$ 2.9.ad_n. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.9.ad_n : 4.0.10933.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 2.27.an_dl. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.27.an_dl : 4.0.10933.1.

## 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 4.3.af_i_g_abh $2$ (not in LMFDB) 4.3.ab_ae_a_v $2$ (not in LMFDB) 4.3.b_ae_a_v $2$ (not in LMFDB) 4.3.f_i_ag_abh $2$ (not in LMFDB) 4.3.h_u_be_bn $2$ (not in LMFDB) 4.3.ae_i_ap_be $3$ (not in LMFDB) 4.3.ab_f_aj_m $3$ (not in LMFDB) 4.3.c_c_ad_ag $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.af_i_g_abh $2$ (not in LMFDB) 4.3.ab_ae_a_v $2$ (not in LMFDB) 4.3.b_ae_a_v $2$ (not in LMFDB) 4.3.f_i_ag_abh $2$ (not in LMFDB) 4.3.h_u_be_bn $2$ (not in LMFDB) 4.3.ae_i_ap_be $3$ (not in LMFDB) 4.3.ab_f_aj_m $3$ (not in LMFDB) 4.3.c_c_ad_ag $3$ (not in LMFDB) 4.3.ab_c_ag_p $4$ (not in LMFDB) 4.3.b_c_g_p $4$ (not in LMFDB) 4.3.ac_c_d_ag $6$ (not in LMFDB) 4.3.b_f_j_m $6$ (not in LMFDB) 4.3.e_i_p_be $6$ (not in LMFDB) 4.3.ab_ah_d_y $12$ (not in LMFDB) 4.3.ab_c_ag_p $12$ (not in LMFDB) 4.3.b_ah_ad_y $12$ (not in LMFDB) 4.3.ab_ab_ad_s $24$ (not in LMFDB) 4.3.b_ab_d_s $24$ (not in LMFDB)