# Properties

 Label 4.3.ai_bf_adb_fx 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: $( 1 - 3 x + 3 x^{2} )( 1 - 5 x + 13 x^{2} - 25 x^{3} + 39 x^{4} - 45 x^{5} + 27 x^{6} )$ Frobenius angles: $\pm0.0714477711956$, $\pm0.166666666667$, $\pm0.272071776080$, $\pm0.560185743604$ 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 $A(\F_{q^r})$ 5 5425 457940 44078125 4326589525 300106399600 21852060841280 1845111414453125 152838760198077380 12202985346955860625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -4 8 23 84 296 773 2089 6532 20039 59268

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 3.3.af_n_az 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.cc $\times$ 3.729.al_cmn_aoxb. The endomorphism algebra for each factor is: 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 3.729.al_cmn_aoxb : 6.0.1342367.1.
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 $\times$ 3.9.b_ad_ah. 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 $\times$ 3.27.af_h_ev. 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.c_b_b_d $2$ (not in LMFDB) 4.3.i_bf_db_fx $2$ (not in LMFDB) 4.3.af_q_abo_da $3$ (not in LMFDB) 4.3.ac_b_ab_d $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.c_b_b_d $2$ (not in LMFDB) 4.3.i_bf_db_fx $2$ (not in LMFDB) 4.3.af_q_abo_da $3$ (not in LMFDB) 4.3.ac_b_ab_d $3$ (not in LMFDB) 4.3.af_q_abo_da $6$ (not in LMFDB) 4.3.f_q_bo_da $6$ (not in LMFDB)