# Properties

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

## Invariants

 Base field: $\F_{3}$ Dimension: $4$ L-polynomial: $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )( 1 - 3 x + 5 x^{2} - 9 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0975263560046$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.527857038681$ Angle rank: $3$ (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 $A(\F_{q^r})$ 6 6804 584136 43164576 4095464736 310007984832 22788608466462 1866925407215232 155128934657036088 12372374984414505984

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 10 29 82 281 799 2180 6610 20333 60085

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 1.3.ac $\times$ 2.3.ad_f 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 $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.cc $\times$ 2.729.cj_ddt. 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.cj_ddt : 4.0.2197.1.
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 $\times$ 1.9.c $\times$ 2.9.b_al. 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.aj_ct. The endomorphism algebra for each factor is:

## 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.ae_i_ar_bk $2$ (not in LMFDB) 4.3.ac_c_ab_a $2$ (not in LMFDB) 4.3.ac_c_f_am $2$ (not in LMFDB) 4.3.c_c_af_am $2$ (not in LMFDB) 4.3.c_c_b_a $2$ (not in LMFDB) 4.3.e_i_r_bk $2$ (not in LMFDB) 4.3.i_bg_dh_gm $2$ (not in LMFDB) 4.3.af_r_abr_dg $3$ (not in LMFDB) 4.3.ac_c_ab_a $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ae_i_ar_bk $2$ (not in LMFDB) 4.3.ac_c_ab_a $2$ (not in LMFDB) 4.3.ac_c_f_am $2$ (not in LMFDB) 4.3.c_c_af_am $2$ (not in LMFDB) 4.3.c_c_b_a $2$ (not in LMFDB) 4.3.e_i_r_bk $2$ (not in LMFDB) 4.3.i_bg_dh_gm $2$ (not in LMFDB) 4.3.af_r_abr_dg $3$ (not in LMFDB) 4.3.ac_c_ab_a $3$ (not in LMFDB) 4.3.ab_f_al_m $6$ (not in LMFDB) 4.3.b_f_l_m $6$ (not in LMFDB) 4.3.f_r_br_dg $6$ (not in LMFDB)