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

• Label: 4.3.ai_bc_aci_eb
• 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:	$( 1 - 3 x + 3 x^{2} )^{2}( 1 - 2 x + x^{2} - 6 x^{3} + 9 x^{4} )$
	$1 - 8 x + 28 x^{2} - 60 x^{3} + 105 x^{4} - 180 x^{5} + 252 x^{6} - 216 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.0292466093486$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.637420057318$
Angle rank:	$1$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3$	$2793$	$254016$	$45785649$	$4283299443$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$2$	$8$	$86$	$296$	$746$	$2264$	$6758$	$19304$	$58082$

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_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^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• 2.3.ac_b : \(\Q(\sqrt{-2}, \sqrt{-3})\).

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu^2 $\times$ 1.729.cc^2. The endomorphism algebra for each factor is:

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

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

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

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
4.3.ae_e_m_abn	$2$	(not in LMFDB)
4.3.ac_ac_a_p	$2$	(not in LMFDB)
4.3.c_ac_a_p	$2$	(not in LMFDB)
4.3.e_e_am_abn	$2$	(not in LMFDB)
4.3.i_bc_ci_eb	$2$	(not in LMFDB)
4.3.af_n_abe_ci	$3$	(not in LMFDB)
4.3.ac_ac_a_p	$3$	(not in LMFDB)
4.3.ac_b_ag_y	$3$	(not in LMFDB)
4.3.ac_h_as_y	$3$	(not in LMFDB)
4.3.b_b_ag_am	$3$	(not in LMFDB)
4.3.b_e_ad_g	$3$	(not in LMFDB)
4.3.e_e_am_abn	$3$	(not in LMFDB)
4.3.e_h_a_am	$3$	(not in LMFDB)
4.3.e_q_bk_da	$3$	(not in LMFDB)
4.3.h_bc_cx_fu	$3$	(not in LMFDB)
4.3.k_bx_fu_ma	$3$	(not in LMFDB)