# Properties

 Label 3.3.aj_bk_add Base Field $\F_{3}$ Dimension $3$ Ordinary No $p$-rank $0$ Principally polarizable Yes Contains a Jacobian No

## Invariants

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

This isogeny class is not simple.

## Newton polygon

This isogeny class is supersingular.

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

## 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 $A(\F_{q^r})$ 1 343 21952 753571 19902511 481890304 11681631109 293151929707 7626759805504 203370086883943

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -5 1 28 109 325 892 2431 6805 19684 58321

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 3 and its endomorphism algebra is $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over $$\Q$$ ramified at $3$ and $\infty$.
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 3 and its endomorphism algebra is $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 3 and its endomorphism algebra is $\mathrm{M}_{3}($$$\Q(\sqrt{-3})$$$)$

## 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.ad_a_j $2$ 3.9.aj_cc_ahh 3.3.d_a_aj $2$ 3.9.aj_cc_ahh 3.3.j_bk_dd $2$ 3.9.aj_cc_ahh 3.3.ag_s_abk $3$ (not in LMFDB) 3.3.ad_a_j $3$ (not in LMFDB) 3.3.ad_j_as $3$ (not in LMFDB) 3.3.a_a_a $3$ (not in LMFDB) 3.3.a_j_a $3$ (not in LMFDB) 3.3.d_a_aj $3$ (not in LMFDB) 3.3.d_j_s $3$ (not in LMFDB) 3.3.g_s_bk $3$ (not in LMFDB) 3.3.j_bk_dd $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 3.3.ad_a_j $2$ 3.9.aj_cc_ahh 3.3.d_a_aj $2$ 3.9.aj_cc_ahh 3.3.j_bk_dd $2$ 3.9.aj_cc_ahh 3.3.ag_s_abk $3$ (not in LMFDB) 3.3.ad_a_j $3$ (not in LMFDB) 3.3.ad_j_as $3$ (not in LMFDB) 3.3.a_a_a $3$ (not in LMFDB) 3.3.a_j_a $3$ (not in LMFDB) 3.3.d_a_aj $3$ (not in LMFDB) 3.3.d_j_s $3$ (not in LMFDB) 3.3.g_s_bk $3$ (not in LMFDB) 3.3.j_bk_dd $3$ (not in LMFDB) 3.3.ad_g_aj $4$ (not in LMFDB) 3.3.d_g_j $4$ (not in LMFDB) 3.3.a_a_aj $9$ (not in LMFDB) 3.3.a_a_j $9$ (not in LMFDB) 3.3.ad_ad_s $12$ (not in LMFDB) 3.3.a_ad_a $12$ (not in LMFDB) 3.3.a_g_a $12$ (not in LMFDB) 3.3.d_ad_as $12$ (not in LMFDB) 3.3.ad_d_a $24$ (not in LMFDB) 3.3.a_d_a $24$ (not in LMFDB) 3.3.d_d_a $24$ (not in LMFDB)