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

• Label: 5.3.al_cg_aho_sp_abjx
• 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:	$( 1 - 3 x + 3 x^{2} )^{2}( 1 - 5 x + 13 x^{2} - 25 x^{3} + 39 x^{4} - 45 x^{5} + 27 x^{6} )$
	$1 - 11 x + 58 x^{2} - 196 x^{3} + 483 x^{4} - 933 x^{5} + 1449 x^{6} - 1764 x^{7} + 1566 x^{8} - 891 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.0714477711956$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.272071776080$, $\pm0.560185743604$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5$	$37975$	$12822320$	$4011109375$	$1172505761275$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-7$	$5$	$23$	$93$	$323$	$827$	$2170$	$6613$	$20039$	$59025$

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.af_n_az 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.af_n_az : 6.0.1342367.1.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc^2 $\times$ 3.729.al_cmn_aoxb. 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.al_cmn_aoxb : 6.0.1342367.1.

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.b_ad_ah. The endomorphism algebra for each factor is:

  • 1.9.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 3.9.b_ad_ah : 6.0.1342367.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^2 $\times$ 3.27.af_h_ev. The endomorphism algebra for each factor is:

  • 1.27.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 3.27.af_h_ev : 6.0.1342367.1.

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.ab_ac_e_d_ad	$2$	(not in LMFDB)
5.3.f_k_k_j_p	$2$	(not in LMFDB)
5.3.l_cg_ho_sp_bjx	$2$	(not in LMFDB)
5.3.ai_bi_adz_jm_asg	$3$	(not in LMFDB)
5.3.af_k_ak_j_ap	$3$	(not in LMFDB)
5.3.af_t_acd_ew_ajg	$3$	(not in LMFDB)
5.3.ac_e_ah_g_ag	$3$	(not in LMFDB)
5.3.b_ac_ae_d_d	$3$	(not in LMFDB)