# Properties

 Label 4.3.ah_ba_aco_ez Base Field $\F_{3}$ Dimension $4$ Ordinary No $p$-rank $2$ Principally polarizable Yes Contains a Jacobian No

## Invariants

 Base field: $\F_{3}$ Dimension: $4$ L-polynomial: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - x + 5 x^{2} - 3 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.345303779071$, $\pm0.557095674046$ Angle rank: $2$ (numerical)

This isogeny class is not simple.

## Newton polygon

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

## 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 $A(\F_{q^r})$ 11 10241 681296 50191141 4407635056 314606141696 23385096417001 1928181068114301 154310471689556336 12051248646952798976

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 13 33 93 302 811 2237 6821 20229 58528

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 2.3.ab_f 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_f : 4.0.2725.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 $\times$ 2.729.abb_ef. 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.abb_ef : 4.0.2725.1.
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.j_bl. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.9.j_bl : 4.0.2725.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 2.27.f_ab. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.27.f_ab : 4.0.2725.1.

## 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_o_abe_cf $2$ (not in LMFDB) 4.3.ab_c_a_d $2$ (not in LMFDB) 4.3.b_c_a_d $2$ (not in LMFDB) 4.3.f_o_be_cf $2$ (not in LMFDB) 4.3.h_ba_co_ez $2$ (not in LMFDB) 4.3.ae_o_abh_co $3$ (not in LMFDB) 4.3.ab_c_a_d $3$ (not in LMFDB) 4.3.ab_l_aj_bw $3$ (not in LMFDB) 4.3.c_i_p_be $3$ (not in LMFDB) 4.3.f_o_be_cf $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.af_o_abe_cf $2$ (not in LMFDB) 4.3.ab_c_a_d $2$ (not in LMFDB) 4.3.b_c_a_d $2$ (not in LMFDB) 4.3.f_o_be_cf $2$ (not in LMFDB) 4.3.h_ba_co_ez $2$ (not in LMFDB) 4.3.ae_o_abh_co $3$ (not in LMFDB) 4.3.ab_c_a_d $3$ (not in LMFDB) 4.3.ab_l_aj_bw $3$ (not in LMFDB) 4.3.c_i_p_be $3$ (not in LMFDB) 4.3.f_o_be_cf $3$ (not in LMFDB) 4.3.ab_i_ag_bh $4$ (not in LMFDB) 4.3.b_i_g_bh $4$ (not in LMFDB) 4.3.ac_i_ap_be $6$ (not in LMFDB) 4.3.b_l_j_bw $6$ (not in LMFDB) 4.3.e_o_bh_co $6$ (not in LMFDB) 4.3.ab_ab_d_am $12$ (not in LMFDB) 4.3.b_ab_ad_am $12$ (not in LMFDB) 4.3.ab_f_ad_s $24$ (not in LMFDB) 4.3.b_f_d_s $24$ (not in LMFDB)