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

• Label: 5.3.ak_bu_aez_kb_arr
• 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:	$( 1 - 3 x + 3 x^{2} )^{3}( 1 - x + x^{2} - 3 x^{3} + 9 x^{4} )$
	$1 - 10 x + 46 x^{2} - 129 x^{3} + 261 x^{4} - 459 x^{5} + 783 x^{6} - 1161 x^{7} + 1242 x^{8} - 810 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.201748855633$, $\pm0.672988571819$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7$	$36015$	$12139456$	$6725621175$	$1406550257392$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$2$	$21$	$134$	$369$	$881$	$2514$	$6806$	$19173$	$58157$

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_b 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_b : 4.0.16317.1.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc^3 $\times$ 2.729.al_abfv. 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.al_abfv : 4.0.16317.1.

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.b_n. The endomorphism algebra for each factor is:

  • 1.9.ad^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
  • 2.9.b_n : 4.0.16317.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^3 $\times$ 2.27.ah_t. The endomorphism algebra for each factor is:

  • 1.27.a^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
  • 2.27.ah_t : 4.0.16317.1.

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_bc_abz_bt_abb	$2$	(not in LMFDB)
5.3.ae_e_d_j_abt	$2$	(not in LMFDB)
5.3.ac_ac_j_j_abt	$2$	(not in LMFDB)
5.3.c_ac_aj_j_bt	$2$	(not in LMFDB)
5.3.e_e_ad_j_bt	$2$	(not in LMFDB)
5.3.i_bc_bz_bt_bb	$2$	(not in LMFDB)
5.3.k_bu_ez_kb_rr	$2$	(not in LMFDB)
5.3.ah_z_acl_ff_ajs	$3$	(not in LMFDB)
5.3.ae_e_d_j_abt	$3$	(not in LMFDB)
5.3.ae_n_abh_cu_aew	$3$	(not in LMFDB)
5.3.ab_b_ad_j_a	$3$	(not in LMFDB)
5.3.ab_k_am_bt_acc	$3$	(not in LMFDB)
5.3.c_ac_aj_j_bt	$3$	(not in LMFDB)
5.3.c_h_j_s_s	$3$	(not in LMFDB)
5.3.f_n_v_bb_bk	$3$	(not in LMFDB)
5.3.i_bc_bz_bt_bb	$3$	(not in LMFDB)