# Properties

 Label 3.3.ag_r_abh Base Field $\F_{3}$ Dimension $3$ Ordinary No $p$-rank $2$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $3$ L-polynomial: $( 1 - 3 x + 3 x^{2} )( 1 - 3 x + 5 x^{2} - 9 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.0975263560046$, $\pm0.166666666667$, $\pm0.527857038681$ Angle rank: $2$ (numerical) Jacobians: 0

This isogeny class is not simple.

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

## 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 $A(\F_{q^r})$ 3 567 15372 449631 16923408 453228048 10841393181 285987347919 7805622152412 207827302701312

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -2 8 19 68 283 845 2266 6644 20143 59603

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad $\times$ 2.3.ad_f 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.cj_ddt. The endomorphism algebra for each factor is: 1.729.cc : the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$. 2.729.cj_ddt : 4.0.2197.1.
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.b_al. 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.aj_ct. 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.a_ab_d $2$ 3.9.ac_af_bz 3.3.g_r_bh $2$ 3.9.ac_af_bz 3.3.ad_i_as $3$ (not in LMFDB) 3.3.a_ab_ad $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.a_ab_d $2$ 3.9.ac_af_bz 3.3.g_r_bh $2$ 3.9.ac_af_bz 3.3.ad_i_as $3$ (not in LMFDB) 3.3.a_ab_ad $3$ (not in LMFDB) 3.3.ad_i_as $6$ (not in LMFDB) 3.3.d_i_s $6$ (not in LMFDB)