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

• Label: 5.3.am_cq_ajj_xx_abut
• Base field: $\F_{3}$
• Dimension: $5$
• $p$-rank: $2$
• 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:	$( 1 - 3 x + 3 x^{2} )^{3}( 1 - 3 x + 5 x^{2} - 9 x^{3} + 9 x^{4} )$
	$1 - 12 x + 68 x^{2} - 243 x^{3} + 621 x^{4} - 1215 x^{5} + 1863 x^{6} - 2187 x^{7} + 1836 x^{8} - 972 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.0975263560046$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.527857038681$
Angle rank:	$2$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3$	$27783$	$12051648$	$3723394311$	$1242872006928$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-8$	$2$	$19$	$86$	$337$	$953$	$2428$	$6806$	$20143$	$59117$

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^3 $\times$ 2.3.ad_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^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
• 2.3.ad_f : 4.0.2197.1.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc^3 $\times$ 2.729.cj_ddt. The endomorphism algebra for each factor is:

• 1.729.cc^3 : $\mathrm{M}_{3}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 2.729.cj_ddt : 4.0.2197.1.

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^3 $\times$ 2.9.b_al. The endomorphism algebra for each factor is:

  • 1.9.ad^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
  • 2.9.b_al : 4.0.2197.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^3 $\times$ 2.27.aj_ct. The endomorphism algebra for each factor is:

  • 1.27.a^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
  • 2.27.aj_ct : 4.0.2197.1.

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.ag_o_aj_abb_dd	$2$	(not in LMFDB)
5.3.a_ae_d_j_aj	$2$	(not in LMFDB)
5.3.g_o_p_j_j	$2$	(not in LMFDB)
5.3.m_cq_jj_xx_but	$2$	(not in LMFDB)
5.3.aj_bp_aez_md_axo	$3$	(not in LMFDB)
5.3.ag_o_ap_j_aj	$3$	(not in LMFDB)
5.3.ag_x_acr_gg_alu	$3$	(not in LMFDB)
5.3.ad_f_aj_j_a	$3$	(not in LMFDB)
5.3.ad_o_abk_dd_agg	$3$	(not in LMFDB)
5.3.a_ae_ad_j_j	$3$	(not in LMFDB)
5.3.a_f_ad_a_as	$3$	(not in LMFDB)
5.3.d_f_d_aj_abk	$3$	(not in LMFDB)
5.3.g_o_j_abb_add	$3$	(not in LMFDB)