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

• Label: 5.3.ak_bw_aft_nh_azb
• 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:	$( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 9 x^{2} - 17 x^{3} + 27 x^{4} - 36 x^{5} + 27 x^{6} )$
	$1 - 10 x + 48 x^{2} - 149 x^{3} + 345 x^{4} - 651 x^{5} + 1035 x^{6} - 1341 x^{7} + 1296 x^{8} - 810 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.0653366913680$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.328985474983$, $\pm0.609104440316$
Angle rank:	$3$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7$	$41503$	$11217472$	$4159306151$	$1059191953232$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$6$	$21$	$94$	$299$	$741$	$2220$	$6950$	$20055$	$58621$

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 $\F_{3^{6}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ad^2 $\times$ 3.3.ae_j_ar 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 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• 3.3.ae_j_ar : 6.0.10338167.1.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc^2 $\times$ 3.729.adt_hiy_ajlhg. 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 $\infty$.
• 3.729.adt_hiy_ajlhg : 6.0.10338167.1.

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 $\times$ 3.9.c_ab_abl. The endomorphism algebra for each factor is:

  • 1.9.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 3.9.c_ab_abl : 6.0.10338167.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^2 $\times$ 3.27.ah_ay_pq. The endomorphism algebra for each factor is:

  • 1.27.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 3.27.ah_ay_pq : 6.0.10338167.1.

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
5.3.ae_g_af_j_av	$2$	(not in LMFDB)
5.3.ac_a_f_ad_d	$2$	(not in LMFDB)
5.3.c_a_af_ad_ad	$2$	(not in LMFDB)
5.3.e_g_f_j_v	$2$	(not in LMFDB)
5.3.k_bw_ft_nh_zb	$2$	(not in LMFDB)
5.3.ah_bb_acz_gv_amy	$3$	(not in LMFDB)
5.3.ae_g_af_j_av	$3$	(not in LMFDB)
5.3.ae_p_abp_dm_ags	$3$	(not in LMFDB)
5.3.ab_d_af_d_am	$3$	(not in LMFDB)
5.3.c_a_af_ad_ad	$3$	(not in LMFDB)