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

• Label: 5.3.ak_bx_agd_ox_abct
• 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:	$( 1 - 3 x + 3 x^{2} )^{2}( 1 - 4 x + 10 x^{2} - 21 x^{3} + 30 x^{4} - 36 x^{5} + 27 x^{6} )$
	$1 - 10 x + 49 x^{2} - 159 x^{3} + 387 x^{4} - 747 x^{5} + 1161 x^{6} - 1431 x^{7} + 1323 x^{8} - 810 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.0145064862012$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.383559653096$, $\pm0.564732805964$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$7$	$44247$	$11398576$	$2968663971$	$907636535197$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$8$	$21$	$68$	$264$	$773$	$2220$	$6740$	$19740$	$57488$

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_k_av 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_k_av : 6.0.309123.1.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc^2 $\times$ 3.729.acn_djq_adrer. 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.acn_djq_adrer : 6.0.309123.1.

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.e_ai_acx. The endomorphism algebra for each factor is:

  • 1.9.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 3.9.e_ai_acx : 6.0.309123.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^2 $\times$ 3.27.ah_ai_hh. The endomorphism algebra for each factor is:

  • 1.27.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 3.27.ah_ai_hh : 6.0.309123.1.

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.ac_b_d_aj_bb	$2$	(not in LMFDB)
5.3.e_h_j_j_j	$2$	(not in LMFDB)
5.3.k_bx_gd_ox_bct	$2$	(not in LMFDB)
5.3.ah_bc_adg_hq_aoo	$3$	(not in LMFDB)
5.3.ae_b_p_aj_abb	$3$	(not in LMFDB)
5.3.ae_h_aj_j_aj	$3$	(not in LMFDB)
5.3.ae_h_aj_bb_acl	$3$	(not in LMFDB)
5.3.ae_q_abt_dv_ahq	$3$	(not in LMFDB)
5.3.ab_ac_a_a_s	$3$	(not in LMFDB)
5.3.ab_e_ag_a_as	$3$	(not in LMFDB)
5.3.ab_e_ag_s_as	$3$	(not in LMFDB)
5.3.c_af_ap_j_cl	$3$	(not in LMFDB)
5.3.c_b_ad_aj_abb	$3$	(not in LMFDB)
5.3.c_b_ad_j_bb	$3$	(not in LMFDB)
5.3.c_e_d_aj_as	$3$	(not in LMFDB)
5.3.c_k_p_bt_cc	$3$	(not in LMFDB)
5.3.f_k_g_as_acc	$3$	(not in LMFDB)
5.3.f_q_bk_cu_ew	$3$	(not in LMFDB)
5.3.i_z_bb_abt_agp	$3$	(not in LMFDB)
5.3.i_bf_cx_ff_ir	$3$	(not in LMFDB)