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

• Label: 5.3.ak_bv_afj_lr_avj
• 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:	$( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 8 x^{2} - 13 x^{3} + 24 x^{4} - 36 x^{5} + 27 x^{6} )$
	$1 - 10 x + 47 x^{2} - 139 x^{3} + 303 x^{4} - 555 x^{5} + 909 x^{6} - 1251 x^{7} + 1269 x^{8} - 810 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.102762435325$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.278353759721$, $\pm0.643265352440$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7$	$38759$	$11464432$	$5430174659$	$1229650350257$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$4$	$21$	$116$	$334$	$781$	$2318$	$6884$	$19740$	$59324$

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_i_an 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_i_an : 6.0.10816643.1.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc^2 $\times$ 3.729.acf_dba_adgnl. 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.acf_dba_adgnl : 6.0.10816643.1.

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.a_i_at. The endomorphism algebra for each factor is:

  • 1.9.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 3.9.a_i_at : 6.0.10816643.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^2 $\times$ 3.27.ah_ae_gf. The endomorphism algebra for each factor is:

  • 1.27.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 3.27.ah_ae_gf : 6.0.10816643.1.

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_f_ab_j_abh	$2$	(not in LMFDB)
5.3.ac_ab_h_d_av	$2$	(not in LMFDB)
5.3.c_ab_ah_d_v	$2$	(not in LMFDB)
5.3.e_f_b_j_bh	$2$	(not in LMFDB)
5.3.k_bv_fj_lr_vj	$2$	(not in LMFDB)
5.3.ah_ba_acs_ga_ali	$3$	(not in LMFDB)
5.3.ae_o_abl_dd_afu	$3$	(not in LMFDB)
5.3.ab_c_ae_g_ag	$3$	(not in LMFDB)