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

• Label: 5.3.al_cg_ahn_sj_abji
• 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:	$( 1 - 3 x + 3 x^{2} )^{3}( 1 - 2 x + 4 x^{2} - 6 x^{3} + 9 x^{4} )$
	$1 - 11 x + 58 x^{2} - 195 x^{3} + 477 x^{4} - 918 x^{5} + 1431 x^{6} - 1755 x^{7} + 1566 x^{8} - 891 x^{9} + 243 x^{10}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.210767374595$, $\pm0.567777800232$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6$	$45276$	$15410304$	$5172511344$	$1483254534786$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-7$	$5$	$26$	$113$	$383$	$944$	$2345$	$6737$	$19682$	$57605$

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.ac_e 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.ac_e : 4.0.7488.1.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc^3 $\times$ 2.729.ca_czy. 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.ca_czy : 4.0.7488.1.

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.e_k. The endomorphism algebra for each factor is:

  • 1.9.ad^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
  • 2.9.e_k : 4.0.7488.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^3 $\times$ 2.27.ac_bc. The endomorphism algebra for each factor is:

  • 1.27.a^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
  • 2.27.ac_bc : 4.0.7488.1.

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.ah_w_abn_bt_acc	$2$	(not in LMFDB)
5.3.af_k_aj_j_as	$2$	(not in LMFDB)
5.3.ab_ac_d_j_as	$2$	(not in LMFDB)
5.3.b_ac_ad_j_s	$2$	(not in LMFDB)
5.3.f_k_j_j_s	$2$	(not in LMFDB)
5.3.h_w_bn_bt_cc	$2$	(not in LMFDB)
5.3.l_cg_hn_sj_bji	$2$	(not in LMFDB)
5.3.ai_bi_ady_jj_asa	$3$	(not in LMFDB)
5.3.af_k_aj_j_as	$3$	(not in LMFDB)
5.3.af_t_acc_ew_aja	$3$	(not in LMFDB)
5.3.ac_e_ag_j_a	$3$	(not in LMFDB)
5.3.ac_n_ay_cu_aee	$3$	(not in LMFDB)
5.3.b_ac_ad_j_s	$3$	(not in LMFDB)
5.3.b_h_g_s_s	$3$	(not in LMFDB)
5.3.e_k_s_bb_bk	$3$	(not in LMFDB)
5.3.h_w_bn_bt_cc	$3$	(not in LMFDB)