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

• Label: 5.3.al_cf_ahe_qz_abgf
• Base field: $\F_{3}$
• Dimension: $5$
• $p$-rank: $1$
• 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:	$( 1 + x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{4}$
	$1 - 11 x + 57 x^{2} - 186 x^{3} + 441 x^{4} - 837 x^{5} + 1323 x^{6} - 1674 x^{7} + 1539 x^{8} - 891 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.593214749339$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$5$	$36015$	$12293120$	$5143122075$	$1483234632275$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-7$	$3$	$20$	$111$	$383$	$936$	$2429$	$6999$	$19820$	$57603$

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^4 $\times$ 1.3.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^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-3}) \)$)$
• 1.3.b : \(\Q(\sqrt{-11}) \).

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

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

• 1.729.ak : \(\Q(\sqrt{-11}) \).
• 1.729.cc^4 : $\mathrm{M}_{4}(B)$, 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^4 $\times$ 1.9.f. The endomorphism algebra for each factor is:

  • 1.9.ad^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-3}) \)$)$
  • 1.9.f : \(\Q(\sqrt{-11}) \).
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.ai $\times$ 1.27.a^4. The endomorphism algebra for each factor is:

  • 1.27.ai : \(\Q(\sqrt{-11}) \).
  • 1.27.a^4 : $\mathrm{M}_{4}($\(\Q(\sqrt{-3}) \)$)$

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.an_dd_amg_bhp_acpn	$2$	(not in LMFDB)
5.3.ah_v_abe_j_bb	$2$	(not in LMFDB)
5.3.af_j_ag_j_abb	$2$	(not in LMFDB)
5.3.ab_ad_g_j_abb	$2$	(not in LMFDB)
5.3.b_ad_ag_j_bb	$2$	(not in LMFDB)
5.3.f_j_g_j_bb	$2$	(not in LMFDB)
5.3.h_v_be_j_abb	$2$	(not in LMFDB)
5.3.l_cf_he_qz_bgf	$2$	(not in LMFDB)
5.3.n_dd_mg_bhp_cpn	$2$	(not in LMFDB)
5.3.ai_bh_ads_ir_aqq	$3$	(not in LMFDB)
5.3.af_j_ag_j_abb	$3$	(not in LMFDB)
5.3.af_s_abz_en_aii	$3$	(not in LMFDB)
5.3.ac_d_ag_j_a	$3$	(not in LMFDB)
5.3.ac_m_ay_cl_aee	$3$	(not in LMFDB)
5.3.b_ad_ag_j_bb	$3$	(not in LMFDB)
5.3.b_g_d_j_a	$3$	(not in LMFDB)
5.3.b_p_m_dm_cc	$3$	(not in LMFDB)
5.3.e_j_m_j_a	$3$	(not in LMFDB)
5.3.e_s_bw_en_ii	$3$	(not in LMFDB)
5.3.h_v_be_j_abb	$3$	(not in LMFDB)
5.3.h_be_dp_ir_qq	$3$	(not in LMFDB)
5.3.k_bz_gs_qz_bhg	$3$	(not in LMFDB)
5.3.n_dd_mg_bhp_cpn	$3$	(not in LMFDB)