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

• Label: 4.3.ah_z_acj_en
• 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:	$( 1 - 3 x + 3 x^{2} )( 1 - 4 x + 10 x^{2} - 19 x^{3} + 30 x^{4} - 36 x^{5} + 27 x^{6} )$
	$1 - 7 x + 25 x^{2} - 61 x^{3} + 117 x^{4} - 183 x^{5} + 225 x^{6} - 189 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.125412673718$, $\pm0.166666666667$, $\pm0.335294135736$, $\pm0.584823404300$
Angle rank:	$3$ (numerical)
Isomorphism classes:	2

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9$	$8001$	$531468$	$49614201$	$4332861549$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$11$	$27$	$91$	$297$	$767$	$2223$	$6979$	$20412$	$58931$

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 $\times$ 3.3.ae_k_at 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 : \(\Q(\sqrt{-3}) \).
• 3.3.ae_k_at : 6.0.11822771.1.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc $\times$ 3.729.ar_fe_abgaf. The endomorphism algebra for each factor is:

• 1.729.cc : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 3.729.ar_fe_abgaf : 6.0.11822771.1.

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 $\times$ 3.9.e_i_f. The endomorphism algebra for each factor is:

  • 1.9.ad : \(\Q(\sqrt{-3}) \).
  • 3.9.e_i_f : 6.0.11822771.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 3.27.ab_ai_jr. The endomorphism algebra for each factor is:

  • 1.27.a : \(\Q(\sqrt{-3}) \).
  • 3.27.ab_ai_jr : 6.0.11822771.1.

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.ab_b_ab_d	$2$	(not in LMFDB)
4.3.b_b_b_d	$2$	(not in LMFDB)
4.3.h_z_cj_en	$2$	(not in LMFDB)
4.3.ae_n_abf_ci	$3$	(not in LMFDB)
4.3.ab_b_ab_d	$3$	(not in LMFDB)