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

• Label: 5.3.al_cl_ajc_yk_abws
• Base field: $\F_{3}$
• Dimension: $5$
• $p$-rank: $4$
• 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:	$( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )^{4}$
	$1 - 11 x + 63 x^{2} - 236 x^{3} + 634 x^{4} - 1266 x^{5} + 1902 x^{6} - 2124 x^{7} + 1701 x^{8} - 891 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.304086723985$, $\pm0.304086723985$
Angle rank:	$1$ (numerical)

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

Newton polygon

$p$-rank:	$4$
Slopes:	$[0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$16$	$145152$	$58383808$	$7729053696$	$929460108016$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-7$	$15$	$68$	$147$	$263$	$600$	$1925$	$6507$	$20444$	$60735$

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

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

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

• 1.729.abu^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-2}) \)$)$
• 1.729.cc : 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 $\times$ 1.9.c^4. The endomorphism algebra for each factor is:

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

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

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.ah_bb_acq_fe_aja	$2$	(not in LMFDB)
5.3.af_p_au_k_s	$2$	(not in LMFDB)
5.3.ad_h_am_bi_aco	$2$	(not in LMFDB)
5.3.ab_d_e_o_ag	$2$	(not in LMFDB)
5.3.b_d_ae_o_g	$2$	(not in LMFDB)
5.3.d_h_m_bi_co	$2$	(not in LMFDB)
5.3.f_p_u_k_as	$2$	(not in LMFDB)
5.3.h_bb_cq_fe_ja	$2$	(not in LMFDB)
5.3.l_cl_jc_yk_bws	$2$	(not in LMFDB)
5.3.ai_bn_aey_mk_aya	$3$	(not in LMFDB)
5.3.af_m_af_abp_eq	$3$	(not in LMFDB)
5.3.af_p_au_k_s	$3$	(not in LMFDB)
5.3.ac_g_e_al_ci	$3$	(not in LMFDB)
5.3.b_ad_k_n_abh	$3$	(not in LMFDB)
5.3.b_a_n_t_a	$3$	(not in LMFDB)
5.3.e_j_bc_cj_ds	$3$	(not in LMFDB)
5.3.h_v_bu_ef_ir	$3$	(not in LMFDB)