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

• Label: 5.3.ak_bt_aeq_ir_aoo
• Base field: $\F_{3}$
• Dimension: $5$
• $p$-rank: $1$
• 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:	$( 1 + 2 x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{4}$
	$1 - 10 x + 45 x^{2} - 120 x^{3} + 225 x^{4} - 378 x^{5} + 675 x^{6} - 1080 x^{7} + 1215 x^{8} - 810 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.695913276015$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6$	$28812$	$11063808$	$6583196256$	$1326820798326$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$0$	$18$	$132$	$354$	$900$	$2598$	$6852$	$19494$	$58560$

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^4 $\times$ 1.3.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^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-3}) \)$)$
• 1.3.c : \(\Q(\sqrt{-2}) \).

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.cc^4. The endomorphism algebra for each factor is:

• 1.729.abu : \(\Q(\sqrt{-2}) \).
• 1.729.cc^4 : $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.

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^4 $\times$ 1.9.c. The endomorphism algebra for each factor is:

  • 1.9.ad^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-3}) \)$)$
  • 1.9.c : \(\Q(\sqrt{-2}) \).
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.ak $\times$ 1.27.a^4. The endomorphism algebra for each factor is:

  • 1.27.ak : \(\Q(\sqrt{-2}) \).
  • 1.27.a^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-3}) \)$)$

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.ao_dp_aou_bpx_adhe	$2$	(not in LMFDB)
5.3.ai_bb_abq_j_cc	$2$	(not in LMFDB)
5.3.ac_ad_m_j_acc	$2$	(not in LMFDB)
5.3.e_d_ag_j_cc	$2$	(not in LMFDB)
5.3.k_bt_eq_ir_oo	$2$	(not in LMFDB)
5.3.ah_y_acf_en_aii	$3$	(not in LMFDB)
5.3.ae_d_g_j_acc	$3$	(not in LMFDB)
5.3.ae_m_abe_cl_aee	$3$	(not in LMFDB)
5.3.ab_a_ad_j_a	$3$	(not in LMFDB)
5.3.ab_j_am_bk_acc	$3$	(not in LMFDB)
5.3.c_ad_am_j_cc	$3$	(not in LMFDB)
5.3.c_g_g_j_a	$3$	(not in LMFDB)
5.3.c_p_y_dm_ee	$3$	(not in LMFDB)
5.3.f_m_p_j_a	$3$	(not in LMFDB)
5.3.f_v_ci_fo_kk	$3$	(not in LMFDB)
5.3.i_bb_bq_j_acc	$3$	(not in LMFDB)
5.3.i_bk_ek_kt_uu	$3$	(not in LMFDB)
5.3.l_ci_if_vd_bpo	$3$	(not in LMFDB)
5.3.o_dp_ou_bpx_dhe	$3$	(not in LMFDB)