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

• Label: 5.3.ak_by_agj_pp_abed
• 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:	$( 1 - 3 x + 3 x^{2} )^{3}( 1 - x + 5 x^{2} - 3 x^{3} + 9 x^{4} )$
	$1 - 10 x + 50 x^{2} - 165 x^{3} + 405 x^{4} - 783 x^{5} + 1215 x^{6} - 1485 x^{7} + 1350 x^{8} - 810 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.345303779071$, $\pm0.557095674046$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$11$	$71687$	$19076288$	$4567393831$	$1194469100176$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$10$	$33$	$102$	$329$	$865$	$2318$	$6902$	$20229$	$58285$

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_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.ab_f : 4.0.2725.1.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc^3 $\times$ 2.729.abb_ef. 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.abb_ef : 4.0.2725.1.

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.j_bl. The endomorphism algebra for each factor is:

  • 1.9.ad^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
  • 2.9.j_bl : 4.0.2725.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^3 $\times$ 2.27.f_ab. The endomorphism algebra for each factor is:

  • 1.27.a^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
  • 2.27.f_ab : 4.0.2725.1.

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_bg_adj_hh_ann	$2$	(not in LMFDB)
5.3.ac_c_ad_j_aj	$2$	(not in LMFDB)
5.3.e_i_j_j_j	$2$	(not in LMFDB)
5.3.k_by_gj_pp_bed	$2$	(not in LMFDB)
5.3.ah_bd_adj_hz_apg	$3$	(not in LMFDB)
5.3.ae_i_aj_j_aj	$3$	(not in LMFDB)
5.3.ae_r_abt_ee_ahq	$3$	(not in LMFDB)
5.3.ab_f_ad_j_a	$3$	(not in LMFDB)
5.3.ab_o_am_dd_acc	$3$	(not in LMFDB)
5.3.c_c_d_j_j	$3$	(not in LMFDB)
5.3.c_l_v_cc_dm	$3$	(not in LMFDB)
5.3.f_r_bt_dv_gy	$3$	(not in LMFDB)
5.3.i_bg_dj_hh_nn	$3$	(not in LMFDB)