Properties

 Label 4.3.ah_x_abw_dg 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: $( 1 - 3 x + 3 x^{2} )^{2}( 1 - x + 2 x^{2} - 3 x^{3} + 9 x^{4} )$ Frobenius angles: $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.235082516458$, $\pm0.648854628963$ 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 $A(\F_{q^r})$ 8 6272 476672 72077824 5294214808 310927425536 24317685973624 1894394635241472 146275717043144192 12010242165980304512

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ -3 7 24 123 347 802 2321 6707 19176 58327

Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{3}$
 The isogeny class factors as 1.3.ad 2 $\times$ 2.3.ab_c 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_c : 4.0.3757.1.
Endomorphism algebra over $\overline{\F}_{3}$
 The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc 2 $\times$ 2.729.abk_bas. 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.abk_bas : 4.0.3757.1.
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.d_q. The endomorphism algebra for each factor is: 1.9.ad 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.9.d_q : 4.0.3757.1.
• Endomorphism algebra over $\F_{3^{3}}$  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 2.27.ae_ak. The endomorphism algebra for each factor is: 1.27.a 2 : $\mathrm{M}_{2}($$$\Q(\sqrt{-3})$$$)$ 2.27.ae_ak : 4.0.3757.1.

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_l_am_m $2$ (not in LMFDB) 4.3.ab_ab_a_m $2$ (not in LMFDB) 4.3.b_ab_a_m $2$ (not in LMFDB) 4.3.f_l_m_m $2$ (not in LMFDB) 4.3.h_x_bw_dg $2$ (not in LMFDB) 4.3.ae_l_ay_bw $3$ (not in LMFDB) 4.3.ab_ab_a_m $3$ (not in LMFDB) 4.3.ab_i_aj_be $3$ (not in LMFDB) 4.3.c_f_g_m $3$ (not in LMFDB) 4.3.f_l_m_m $3$ (not in LMFDB)
Below is a list of all twists of this isogeny class.
 Twist Extension Degree Common base change 4.3.af_l_am_m $2$ (not in LMFDB) 4.3.ab_ab_a_m $2$ (not in LMFDB) 4.3.b_ab_a_m $2$ (not in LMFDB) 4.3.f_l_m_m $2$ (not in LMFDB) 4.3.h_x_bw_dg $2$ (not in LMFDB) 4.3.ae_l_ay_bw $3$ (not in LMFDB) 4.3.ab_ab_a_m $3$ (not in LMFDB) 4.3.ab_i_aj_be $3$ (not in LMFDB) 4.3.c_f_g_m $3$ (not in LMFDB) 4.3.f_l_m_m $3$ (not in LMFDB) 4.3.ab_f_ag_y $4$ (not in LMFDB) 4.3.b_f_g_y $4$ (not in LMFDB) 4.3.af_l_am_m $6$ (not in LMFDB) 4.3.ae_l_ay_bw $6$ (not in LMFDB) 4.3.ac_f_ag_m $6$ (not in LMFDB) 4.3.ab_ab_a_m $6$ (not in LMFDB) 4.3.ab_i_aj_be $6$ (not in LMFDB) 4.3.b_ab_a_m $6$ (not in LMFDB) 4.3.b_i_j_be $6$ (not in LMFDB) 4.3.c_f_g_m $6$ (not in LMFDB) 4.3.e_l_y_bw $6$ (not in LMFDB) 4.3.ab_ae_d_g $12$ (not in LMFDB) 4.3.ab_f_ag_y $12$ (not in LMFDB) 4.3.b_ae_ad_g $12$ (not in LMFDB) 4.3.b_f_g_y $12$ (not in LMFDB) 4.3.ab_c_ad_s $24$ (not in LMFDB) 4.3.b_c_d_s $24$ (not in LMFDB)