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

• Label: 4.3.ah_y_acc_dv
• 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:	$( 1 - 3 x + 3 x^{2} )^{2}( 1 - x + 3 x^{2} - 3 x^{3} + 9 x^{4} )$
	$1 - 7 x + 24 x^{2} - 54 x^{3} + 99 x^{4} - 162 x^{5} + 216 x^{6} - 189 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.166666666667$, $\pm0.268536328535$, $\pm0.622727850897$
Angle rank:	$2$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$9$	$7497$	$529200$	$67150629$	$5192572464$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$9$	$27$	$117$	$342$	$783$	$2223$	$6741$	$19521$	$58464$

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.ab_d 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.ab_d : 4.0.11661.1.

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

The base change of $A$ to $\F_{3^{6}}$ is 1.729.cc^2 $\times$ 2.729.acd_deb. The endomorphism algebra for each factor is:

• 1.729.cc^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 2.729.acd_deb : 4.0.11661.1.

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^2 $\times$ 2.9.f_v. The endomorphism algebra for each factor is:

  • 1.9.ad^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 2.9.f_v : 4.0.11661.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a^2 $\times$ 2.27.ab_abb. The endomorphism algebra for each factor is:

  • 1.27.a^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 2.27.ab_abb : 4.0.11661.1.

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.af_m_as_bb	$2$	(not in LMFDB)
4.3.ab_a_a_j	$2$	(not in LMFDB)
4.3.b_a_a_j	$2$	(not in LMFDB)
4.3.f_m_s_bb	$2$	(not in LMFDB)
4.3.h_y_cc_dv	$2$	(not in LMFDB)
4.3.ae_m_abb_cc	$3$	(not in LMFDB)
4.3.ab_a_a_j	$3$	(not in LMFDB)
4.3.ab_j_aj_bk	$3$	(not in LMFDB)
4.3.c_g_j_s	$3$	(not in LMFDB)
4.3.f_m_s_bb	$3$	(not in LMFDB)