# Properties

 Label 4.3.ah_x_abx_dj 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: $( 1 - 3 x + 3 x^{2} )( 1 - 4 x + 8 x^{2} - 13 x^{3} + 24 x^{4} - 36 x^{5} + 27 x^{6} )$ Frobenius angles: $\pm0.102762435325$, $\pm0.166666666667$, $\pm0.278353759721$, $\pm0.643265352440$ 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 $A(\F_{q^r})$ 7 5537 409444 59672249 4537455167 281119337072 23392114502071 1922148223120489 150528923543557888 12264624935213421617

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 7 21 107 307 727 2237 6803 19740 59567

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 3.3.ae_i_an 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.acf_dba_adgnl. The endomorphism algebra for each factor is: 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 3.729.acf_dba_adgnl : 6.0.10816643.1.
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.a_i_at. 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.ah_ae_gf. 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_ab_ab_j $2$ (not in LMFDB) 4.3.b_ab_b_j $2$ (not in LMFDB) 4.3.h_x_bx_dj $2$ (not in LMFDB) 4.3.ae_l_az_bw $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.ab_ab_ab_j $2$ (not in LMFDB) 4.3.b_ab_b_j $2$ (not in LMFDB) 4.3.h_x_bx_dj $2$ (not in LMFDB) 4.3.ae_l_az_bw $3$ (not in LMFDB) 4.3.e_l_z_bw $6$ (not in LMFDB)