Abelian variety isogeny class 4.3.ai_bi_adr_hk over $\F_{3}$

• Label: 4.3.ai_bi_adr_hk
• Base field: $\F_{3}$
• Dimension: $4$
• $p$-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:	$4$
L-polynomial:	$( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )( 1 - 3 x + 7 x^{2} - 9 x^{3} + 9 x^{4} )$
	$1 - 8 x + 34 x^{2} - 95 x^{3} + 192 x^{4} - 285 x^{5} + 306 x^{6} - 216 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.227267020856$, $\pm0.304086723985$, $\pm0.464830336654$
Angle rank:	$3$ (numerical)
Isomorphism classes:	1

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10$	$12180$	$1122520$	$57002400$	$4066084000$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$14$	$47$	$106$	$281$	$803$	$2222$	$6370$	$19361$	$59189$

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 $\times$ 2.3.ad_h 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 : \(\Q(\sqrt{-2}) \).
• 2.3.ad_h : 4.0.1525.1.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.cc $\times$ 2.729.cn_dov. The endomorphism algebra for each factor is:

• 1.729.abu : \(\Q(\sqrt{-2}) \).
• 1.729.cc : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 2.729.cn_dov : 4.0.1525.1.

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 $\times$ 1.9.c $\times$ 2.9.f_n. The endomorphism algebra for each factor is:

  • 1.9.ad : \(\Q(\sqrt{-3}) \).
  • 1.9.c : \(\Q(\sqrt{-2}) \).
  • 2.9.f_n : 4.0.1525.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 1.27.k $\times$ 2.27.j_cv. The endomorphism algebra for each factor is:

  • 1.27.a : \(\Q(\sqrt{-3}) \).
  • 1.27.k : \(\Q(\sqrt{-2}) \).
  • 2.27.j_cv : 4.0.1525.1.

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
4.3.ae_k_at_bk	$2$	(not in LMFDB)
4.3.ac_e_af_m	$2$	(not in LMFDB)
4.3.ac_e_b_a	$2$	(not in LMFDB)
4.3.c_e_ab_a	$2$	(not in LMFDB)
4.3.c_e_f_m	$2$	(not in LMFDB)
4.3.e_k_t_bk	$2$	(not in LMFDB)
4.3.i_bi_dr_hk	$2$	(not in LMFDB)
4.3.af_t_abv_ds	$3$	(not in LMFDB)
4.3.ac_e_b_a	$3$	(not in LMFDB)