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

• Label: 4.3.ai_bf_ada_fu
• Base field: $\F_{3}$
• Dimension: $4$
• $p$-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:	$4$
L-polynomial:	$( 1 - 3 x + 3 x^{2} )^{2}( 1 - 2 x + 4 x^{2} - 6 x^{3} + 9 x^{4} )$
	$1 - 8 x + 31 x^{2} - 78 x^{3} + 150 x^{4} - 234 x^{5} + 279 x^{6} - 216 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.166666666667$, $\pm0.210767374595$, $\pm0.567777800232$
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:	$[0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6$	$6468$	$550368$	$56840784$	$5473263966$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$8$	$26$	$104$	$356$	$890$	$2264$	$6656$	$19682$	$57848$

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 $\times$ 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^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• 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^2 $\times$ 2.729.ca_czy. The endomorphism algebra for each factor is:

• 1.729.cc^2 : $\mathrm{M}_{2}(B)$, 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^2 $\times$ 2.9.e_k. The endomorphism algebra for each factor is:

  • 1.9.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 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^2 $\times$ 2.27.ac_bc. The endomorphism algebra for each factor is:

  • 1.27.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 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
4.3.ae_h_ag_g	$2$	(not in LMFDB)
4.3.ac_b_a_g	$2$	(not in LMFDB)
4.3.c_b_a_g	$2$	(not in LMFDB)
4.3.e_h_g_g	$2$	(not in LMFDB)
4.3.i_bf_da_fu	$2$	(not in LMFDB)
4.3.af_q_abn_da	$3$	(not in LMFDB)
4.3.ac_b_a_g	$3$	(not in LMFDB)
4.3.ac_k_as_bq	$3$	(not in LMFDB)
4.3.b_e_d_g	$3$	(not in LMFDB)
4.3.e_h_g_g	$3$	(not in LMFDB)