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

• Label: 4.3.ah_z_ack_eq
• 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:	$( 1 - 3 x + 3 x^{2} )( 1 - 4 x + 10 x^{2} - 20 x^{3} + 30 x^{4} - 36 x^{5} + 27 x^{6} )$
	$1 - 7 x + 25 x^{2} - 62 x^{3} + 120 x^{4} - 186 x^{5} + 225 x^{6} - 189 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.0844416807585$, $\pm0.166666666667$, $\pm0.360432408976$, $\pm0.575465777728$
Angle rank:	$3$ (numerical)
Isomorphism classes:	4

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8$	$7168$	$468608$	$40628224$	$3823530328$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$11$	$24$	$75$	$267$	$746$	$2209$	$6931$	$20328$	$58731$

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_au 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_au : 6.0.2296688.1.

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

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

• 1.729.cc : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 3.729.abm_cpz_adepg : 6.0.2296688.1.

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_a_abi. The endomorphism algebra for each factor is:

  • 1.9.ad : \(\Q(\sqrt{-3}) \).
  • 3.9.e_a_abi : 6.0.2296688.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 3.27.ae_al_ku. The endomorphism algebra for each factor is:

  • 1.27.a : \(\Q(\sqrt{-3}) \).
  • 3.27.ae_al_ku : 6.0.2296688.1.

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_ac_a	$2$	(not in LMFDB)
4.3.b_b_c_a	$2$	(not in LMFDB)
4.3.h_z_ck_eq	$2$	(not in LMFDB)
4.3.ae_n_abg_ci	$3$	(not in LMFDB)