# Properties

 Label 4.3.ah_z_acj_en 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: $( 1 - 3 x + 3 x^{2} )( 1 - 4 x + 10 x^{2} - 19 x^{3} + 30 x^{4} - 36 x^{5} + 27 x^{6} )$ Frobenius angles: $\pm0.125412673718$, $\pm0.166666666667$, $\pm0.335294135736$, $\pm0.584823404300$ 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 $A(\F_{q^r})$ 9 8001 531468 49614201 4332861549 296714332656 23230256952681 1974715900036425 155745282861778176 12133301575821043401

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 11 27 91 297 767 2223 6979 20412 58931

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 3.3.ae_k_at 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.ar_fe_abgaf. The endomorphism algebra for each factor is: 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 3.729.ar_fe_abgaf : 6.0.11822771.1.
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.e_i_f. 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.ab_ai_jr. 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.ab_b_ab_d $2$ (not in LMFDB) 4.3.b_b_b_d $2$ (not in LMFDB) 4.3.h_z_cj_en $2$ (not in LMFDB) 4.3.ae_n_abf_ci $3$ (not in LMFDB) 4.3.ab_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.ab_b_ab_d $2$ (not in LMFDB) 4.3.b_b_b_d $2$ (not in LMFDB) 4.3.h_z_cj_en $2$ (not in LMFDB) 4.3.ae_n_abf_ci $3$ (not in LMFDB) 4.3.ab_b_ab_d $3$ (not in LMFDB) 4.3.ae_n_abf_ci $6$ (not in LMFDB) 4.3.b_b_b_d $6$ (not in LMFDB) 4.3.e_n_bf_ci $6$ (not in LMFDB)