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

• Label: 4.3.ak_bw_afo_ll
• Base field: $\F_{3}$
• Dimension: $4$
• $p$-rank: $1$
• 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:	$( 1 - x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{3}$
	$1 - 10 x + 48 x^{2} - 144 x^{3} + 297 x^{4} - 432 x^{5} + 432 x^{6} - 270 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.406785250661$
Angle rank:	$1$ (numerical)

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3$	$5145$	$790272$	$56517825$	$4239234843$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-6$	$6$	$36$	$102$	$294$	$882$	$2514$	$6918$	$19548$	$57846$

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

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

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

• 1.729.ak : \(\Q(\sqrt{-11}) \).
• 1.729.cc^3 : $\mathrm{M}_{3}(B)$, 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^3 $\times$ 1.9.f. The endomorphism algebra for each factor is:

  • 1.9.ad^3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
  • 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^3 $\times$ 1.27.i. The endomorphism algebra for each factor is:

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

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.ai_be_acu_ff	$2$	(not in LMFDB)
4.3.ac_a_a_j	$2$	(not in LMFDB)
4.3.e_g_a_aj	$2$	(not in LMFDB)
4.3.k_bw_fo_ll	$2$	(not in LMFDB)
4.3.ah_bb_acu_fo	$3$	(not in LMFDB)
4.3.ae_g_a_aj	$3$	(not in LMFDB)
4.3.ae_p_abk_cu	$3$	(not in LMFDB)
4.3.ab_d_a_a	$3$	(not in LMFDB)
4.3.ab_m_aj_cc	$3$	(not in LMFDB)
4.3.c_a_a_j	$3$	(not in LMFDB)
4.3.c_j_s_bk	$3$	(not in LMFDB)
4.3.f_p_bk_cu	$3$	(not in LMFDB)
4.3.i_be_cu_ff	$3$	(not in LMFDB)