# Properties

 Label 5.3.ak_by_agk_pu_abep Base Field $\F_{3}$ Dimension $5$ Ordinary No $p$-rank $4$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $5$ L-polynomial: $( 1 - 3 x + 3 x^{2} )( 1 - 7 x + 26 x^{2} - 67 x^{3} + 131 x^{4} - 201 x^{5} + 234 x^{6} - 189 x^{7} + 81 x^{8} )$ Frobenius angles: $\pm0.0365491909691$, $\pm0.166666666667$, $\pm0.234353293111$, $\pm0.355928212636$, $\pm0.548250857189$ Angle rank: $4$ (numerical)

This isogeny class is not simple.

## Newton polygon $p$-rank: $4$ Slopes: $[0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1]$

## 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 59031 15595524 3463289739 933273474489 209082889676304 47635494725780901 12010765592522223891 2968078619705306842476 709061345815096794379671

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -6 10 30 82 269 742 2080 6482 19776 58315

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 4.3.ah_ba_acp_fb 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 $\times$ 4.729.abq_boh_acfrw_djcin. The endomorphism algebra for each factor is: 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 4.729.abq_boh_acfrw_djcin : 8.0.371495353.1.
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 $\times$ 4.9.d_a_ax_adl. 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 $\times$ 4.27.c_at_ak_ml. 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 5.3.ae_i_ak_i_aj $2$ (not in LMFDB) 5.3.e_i_k_i_j $2$ (not in LMFDB) 5.3.k_by_gk_pu_bep $2$ (not in LMFDB) 5.3.ah_bd_adk_ib_apm $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 5.3.ae_i_ak_i_aj $2$ (not in LMFDB) 5.3.e_i_k_i_j $2$ (not in LMFDB) 5.3.k_by_gk_pu_bep $2$ (not in LMFDB) 5.3.ah_bd_adk_ib_apm $3$ (not in LMFDB) 5.3.e_i_k_i_j $6$ (not in LMFDB) 5.3.h_bd_dk_ib_pm $6$ (not in LMFDB)