Abelian variety isogeny class 5.3.ak_bw_afs_nb_aym over $\F_{3}$

• Label: 5.3.ak_bw_afs_nb_aym
• Base field: $\F_{3}$
• Dimension: $5$
• $p$-rank: $3$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3}$
Dimension:	$5$
L-polynomial:	$( 1 - 2 x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}( 1 - 2 x + 2 x^{2} - 6 x^{3} + 9 x^{4} )$
	$1 - 10 x + 48 x^{2} - 148 x^{3} + 339 x^{4} - 636 x^{5} + 1017 x^{6} - 1332 x^{7} + 1296 x^{8} - 810 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.116139763599$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.616139763599$
Angle rank:	$2$ (numerical)

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8$	$47040$	$12989312$	$5087846400$	$1227099817768$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$6$	$24$	$110$	$334$	$792$	$2290$	$7006$	$20112$	$59046$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{12}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ad^2 $\times$ 1.3.ac $\times$ 2.3.ac_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 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• 1.3.ac : \(\Q(\sqrt{-2}) \).
• 2.3.ac_c : \(\Q(i, \sqrt{5})\).

Endomorphism algebra over $\overline{\F}_{3}$

The base change of $A$ to $\F_{3^{12}}$ is 1.531441.acec^2 $\times$ 1.531441.azi $\times$ 1.531441.sk^2. The endomorphism algebra for each factor is:

• 1.531441.acec^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 1.531441.azi : \(\Q(\sqrt{-2}) \).
• 1.531441.sk^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$

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 $\times$ 1.9.c $\times$ 2.9.a_ac. The endomorphism algebra for each factor is:

  • 1.9.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 1.9.c : \(\Q(\sqrt{-2}) \).
  • 2.9.a_ac : \(\Q(i, \sqrt{5})\).
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^2 $\times$ 1.27.k $\times$ 2.27.ao_du. The endomorphism algebra for each factor is:

  • 1.27.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 1.27.k : \(\Q(\sqrt{-2}) \).
  • 2.27.ao_du : \(\Q(i, \sqrt{5})\).
• Endomorphism algebra over $\F_{3^{4}}$
  The base change of $A$ to $\F_{3^{4}}$ is 1.81.ac^2 $\times$ 1.81.j^2 $\times$ 1.81.o. The endomorphism algebra for each factor is:

  • 1.81.ac^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$
  • 1.81.j^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 1.81.o : \(\Q(\sqrt{-2}) \).
• Endomorphism algebra over $\F_{3^{6}}$
  The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.cc^2 $\times$ 2.729.a_sk. The endomorphism algebra for each factor is:

  • 1.729.abu : \(\Q(\sqrt{-2}) \).
  • 1.729.cc^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
  • 2.729.a_sk : \(\Q(i, \sqrt{5})\).

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
5.3.ag_q_abg_cx_aga	$2$	(not in LMFDB)
5.3.ag_q_aq_av_dg	$2$	(not in LMFDB)
5.3.ae_g_ae_j_ay	$2$	(not in LMFDB)
5.3.ac_a_e_d_am	$2$	(not in LMFDB)
5.3.a_ac_ai_j_y	$2$	(not in LMFDB)
5.3.a_ac_i_j_ay	$2$	(not in LMFDB)
5.3.c_a_ae_d_m	$2$	(not in LMFDB)
5.3.e_g_e_j_y	$2$	(not in LMFDB)
5.3.g_q_q_av_adg	$2$	(not in LMFDB)
5.3.g_q_bg_cx_ga	$2$	(not in LMFDB)
5.3.k_bw_fs_nb_ym	$2$	(not in LMFDB)
5.3.ah_bb_acy_gs_ams	$3$	(not in LMFDB)
5.3.ae_p_abo_dm_agm	$3$	(not in LMFDB)
5.3.ab_d_ae_g_ag	$3$	(not in LMFDB)