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

• Label: 5.3.al_ck_aiv_xl_abuq
• Base field: $\F_{3}$
• Dimension: $5$
• $p$-rank: $3$
• 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:	$( 1 - x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}( 1 - 2 x + 3 x^{2} )^{2}$
	$1 - 11 x + 62 x^{2} - 229 x^{3} + 609 x^{4} - 1212 x^{5} + 1827 x^{6} - 2061 x^{7} + 1674 x^{8} - 891 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.406785250661$
Angle rank:	$2$ (numerical)

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

Newton polygon

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$12$	$105840$	$40755456$	$5723827200$	$916112728212$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-7$	$13$	$56$	$121$	$263$	$736$	$2261$	$6769$	$19928$	$59053$

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 $\times$ 1.3.ac^2 $\times$ 1.3.ab 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 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• 1.3.ac^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
• 1.3.ab : \(\Q(\sqrt{-11}) \).

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu^2 $\times$ 1.729.ak $\times$ 1.729.cc^2. The endomorphism algebra for each factor is:

• 1.729.abu^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
• 1.729.ak : \(\Q(\sqrt{-11}) \).
• 1.729.cc^2 : $\mathrm{M}_{2}(B)$, 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^2 $\times$ 1.9.c^2 $\times$ 1.9.f. The endomorphism algebra for each factor is:

  • 1.9.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 1.9.c^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
  • 1.9.f : \(\Q(\sqrt{-11}) \).
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^2 $\times$ 1.27.i $\times$ 1.27.k^2. The endomorphism algebra for each factor is:

  • 1.27.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 1.27.i : \(\Q(\sqrt{-11}) \).
  • 1.27.k^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$

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.aj_bq_afb_lx_awq	$2$	(not in LMFDB)
5.3.ah_ba_acn_ez_aiu	$2$	(not in LMFDB)
5.3.af_o_abf_cr_afc	$2$	(not in LMFDB)
5.3.af_o_at_j_m	$2$	(not in LMFDB)
5.3.ad_g_an_bh_aci	$2$	(not in LMFDB)
5.3.ad_g_af_j_am	$2$	(not in LMFDB)
5.3.ab_c_al_v_am	$2$	(not in LMFDB)
5.3.ab_c_b_j_am	$2$	(not in LMFDB)
5.3.b_c_ab_j_m	$2$	(not in LMFDB)
5.3.b_c_l_v_m	$2$	(not in LMFDB)
5.3.d_g_f_j_m	$2$	(not in LMFDB)
5.3.d_g_n_bh_ci	$2$	(not in LMFDB)
5.3.f_o_t_j_am	$2$	(not in LMFDB)
5.3.f_o_bf_cr_fc	$2$	(not in LMFDB)
5.3.h_ba_cn_ez_iu	$2$	(not in LMFDB)
5.3.j_bq_fb_lx_wq	$2$	(not in LMFDB)
5.3.l_ck_iv_xl_buq	$2$	(not in LMFDB)
5.3.ai_bm_aeu_lx_axc	$3$	(not in LMFDB)
5.3.af_l_ae_abn_eh	$3$	(not in LMFDB)
5.3.af_o_at_j_m	$3$	(not in LMFDB)
5.3.af_x_acm_gg_ali	$3$	(not in LMFDB)
5.3.ac_f_c_ap_bw	$3$	(not in LMFDB)
5.3.ac_i_ae_p_m	$3$	(not in LMFDB)
5.3.b_ab_i_j_ap	$3$	(not in LMFDB)
5.3.b_c_l_v_m	$3$	(not in LMFDB)
5.3.b_i_r_bb_dg	$3$	(not in LMFDB)
5.3.e_l_bg_cr_eq	$3$	(not in LMFDB)
5.3.h_x_ce_ez_jv	$3$	(not in LMFDB)