# Properties

 Label 3.3.ae_j_ap Base Field $\F_{3}$ Dimension $3$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $3$ L-polynomial: $( 1 - 3 x + 3 x^{2} )( 1 - x + 3 x^{2} - 3 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.268536328535$, $\pm0.622727850897$ Angle rank: $2$ (numerical) Jacobians: 0

This isogeny class is not simple.

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

## 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 1071 18900 737919 19160784 386845200 10239885819 286632090927 7563732995700 204701246781696

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 0 12 27 108 315 729 2142 6660 19521 58707

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 2.3.ab_d 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 $\times$ 2.729.acd_deb. The endomorphism algebra for each factor is: 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 2.729.acd_deb : 4.0.11661.1.
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 $\times$ 2.9.f_v. 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 $\times$ 2.27.ab_abb. 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 3.3.ac_d_ad $2$ 3.9.c_p_bb 3.3.c_d_d $2$ 3.9.c_p_bb 3.3.e_j_p $2$ 3.9.c_p_bb 3.3.ab_g_ag $3$ (not in LMFDB) 3.3.c_d_d $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.ac_d_ad $2$ 3.9.c_p_bb 3.3.c_d_d $2$ 3.9.c_p_bb 3.3.e_j_p $2$ 3.9.c_p_bb 3.3.ab_g_ag $3$ (not in LMFDB) 3.3.c_d_d $3$ (not in LMFDB) 3.3.ac_d_ad $6$ (not in LMFDB) 3.3.ab_g_ag $6$ (not in LMFDB) 3.3.b_g_g $6$ (not in LMFDB)