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

• Label: 4.3.aj_bp_aeq_jm
• 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:	$( 1 - 2 x + 3 x^{2} )( 1 - x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}$
	$1 - 9 x + 41 x^{2} - 120 x^{3} + 246 x^{4} - 360 x^{5} + 369 x^{6} - 243 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.406785250661$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6$	$8820$	$1072512$	$59623200$	$3785589786$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$11$	$46$	$107$	$265$	$782$	$2347$	$6803$	$19738$	$58571$

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 $\times$ 1.3.ac $\times$ 1.3.ab 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 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• 1.3.ac : \(\Q(\sqrt{-2}) \).
• 1.3.ab : \(\Q(\sqrt{-11}) \).

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

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

• 1.729.abu : \(\Q(\sqrt{-2}) \).
• 1.729.ak : \(\Q(\sqrt{-11}) \).
• 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 $\times$ 1.9.c $\times$ 1.9.f. The endomorphism algebra for each factor is:

  • 1.9.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 1.9.c : \(\Q(\sqrt{-2}) \).
  • 1.9.f : \(\Q(\sqrt{-11}) \).
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^2 $\times$ 1.27.i $\times$ 1.27.k. The endomorphism algebra for each factor is:

  • 1.27.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 1.27.i : \(\Q(\sqrt{-11}) \).
  • 1.27.k : \(\Q(\sqrt{-2}) \).

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.ah_z_aci_ek	$2$	(not in LMFDB)
4.3.af_n_ay_bq	$2$	(not in LMFDB)
4.3.ad_f_am_be	$2$	(not in LMFDB)
4.3.ab_b_a_g	$2$	(not in LMFDB)
4.3.b_b_a_g	$2$	(not in LMFDB)
4.3.d_f_a_ag	$2$	(not in LMFDB)
4.3.f_n_y_bq	$2$	(not in LMFDB)
4.3.h_z_ci_ek	$2$	(not in LMFDB)
4.3.j_bp_eq_jm	$2$	(not in LMFDB)
4.3.ag_x_aci_eq	$3$	(not in LMFDB)
4.3.ad_f_a_ag	$3$	(not in LMFDB)
4.3.ad_o_abb_co	$3$	(not in LMFDB)
4.3.a_f_g_m	$3$	(not in LMFDB)
4.3.d_f_m_be	$3$	(not in LMFDB)