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

• Label: 5.3.ak_bv_afi_ll_auu
• 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:	$( 1 - 3 x + 3 x^{2} )^{3}( 1 - x + 2 x^{2} - 3 x^{3} + 9 x^{4} )$
	$1 - 10 x + 47 x^{2} - 138 x^{3} + 297 x^{4} - 540 x^{5} + 891 x^{6} - 1242 x^{7} + 1269 x^{8} - 810 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.235082516458$, $\pm0.648854628963$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8$	$43904$	$13346816$	$6559081984$	$1434732212968$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$4$	$24$	$132$	$374$	$856$	$2402$	$6788$	$19176$	$58084$

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.ab_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^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
• 2.3.ab_c : 4.0.3757.1.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc^3 $\times$ 2.729.abk_bas. 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.abk_bas : 4.0.3757.1.

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.d_q. The endomorphism algebra for each factor is:

  • 1.9.ad^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
  • 2.9.d_q : 4.0.3757.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^3 $\times$ 2.27.ae_ak. The endomorphism algebra for each factor is:

  • 1.27.a^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
  • 2.27.ae_ak : 4.0.3757.1.

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.ai_bd_aci_dd_aee	$2$	(not in LMFDB)
5.3.ae_f_a_j_abk	$2$	(not in LMFDB)
5.3.ac_ab_g_j_abk	$2$	(not in LMFDB)
5.3.c_ab_ag_j_bk	$2$	(not in LMFDB)
5.3.e_f_a_j_bk	$2$	(not in LMFDB)
5.3.i_bd_ci_dd_ee	$2$	(not in LMFDB)
5.3.k_bv_fi_ll_uu	$2$	(not in LMFDB)
5.3.ah_ba_acr_fx_alc	$3$	(not in LMFDB)
5.3.ae_f_a_j_abk	$3$	(not in LMFDB)
5.3.ae_o_abk_dd_afo	$3$	(not in LMFDB)
5.3.ab_c_ad_j_a	$3$	(not in LMFDB)
5.3.ab_l_am_cc_acc	$3$	(not in LMFDB)
5.3.c_ab_ag_j_bk	$3$	(not in LMFDB)
5.3.c_i_m_bb_bk	$3$	(not in LMFDB)
5.3.f_o_bb_bt_cu	$3$	(not in LMFDB)
5.3.i_bd_ci_dd_ee	$3$	(not in LMFDB)