# Properties

 Label 4.3.ah_y_acc_dv Base Field $\F_{3}$ Dimension $4$ Ordinary No $p$-rank $1$ Principally polarizable Yes Contains a Jacobian No

## Invariants

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

This isogeny class is not simple.

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

## 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 7497 529200 67150629 5192572464 303286636800 23234300923311 1904096980028061 148884520287358800 12037866219491196672

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 9 27 117 342 783 2223 6741 19521 58464

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 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: 1.3.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.3.ab_d : 4.0.11661.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 $\times$ 2.729.acd_deb. The endomorphism algebra for each factor is: 1.729.cc 2 : $\mathrm{M}_{2}(B)$, where $B$ is 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 2 $\times$ 2.9.f_v. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.9.f_v : 4.0.11661.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 2.27.ab_abb. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.27.ab_abb : 4.0.11661.1.

## 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.af_m_as_bb $2$ (not in LMFDB) 4.3.ab_a_a_j $2$ (not in LMFDB) 4.3.b_a_a_j $2$ (not in LMFDB) 4.3.f_m_s_bb $2$ (not in LMFDB) 4.3.h_y_cc_dv $2$ (not in LMFDB) 4.3.ae_m_abb_cc $3$ (not in LMFDB) 4.3.ab_a_a_j $3$ (not in LMFDB) 4.3.ab_j_aj_bk $3$ (not in LMFDB) 4.3.c_g_j_s $3$ (not in LMFDB) 4.3.f_m_s_bb $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.af_m_as_bb $2$ (not in LMFDB) 4.3.ab_a_a_j $2$ (not in LMFDB) 4.3.b_a_a_j $2$ (not in LMFDB) 4.3.f_m_s_bb $2$ (not in LMFDB) 4.3.h_y_cc_dv $2$ (not in LMFDB) 4.3.ae_m_abb_cc $3$ (not in LMFDB) 4.3.ab_a_a_j $3$ (not in LMFDB) 4.3.ab_j_aj_bk $3$ (not in LMFDB) 4.3.c_g_j_s $3$ (not in LMFDB) 4.3.f_m_s_bb $3$ (not in LMFDB) 4.3.ab_g_ag_bb $4$ (not in LMFDB) 4.3.b_g_g_bb $4$ (not in LMFDB) 4.3.ac_g_aj_s $6$ (not in LMFDB) 4.3.ab_a_a_j $6$ (not in LMFDB) 4.3.b_j_j_bk $6$ (not in LMFDB) 4.3.e_m_bb_cc $6$ (not in LMFDB) 4.3.ab_ad_d_a $12$ (not in LMFDB) 4.3.b_ad_ad_a $12$ (not in LMFDB) 4.3.ab_d_ad_s $24$ (not in LMFDB) 4.3.b_d_d_s $24$ (not in LMFDB)