Abelian variety isogeny class 3.3.ae_l_av over $\F_{3}$

• Label: 3.3.ae_l_av
• Base field: $\F_{3}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3}$
Dimension:	$3$
L-polynomial:	$( 1 - 3 x + 3 x^{2} )( 1 - x + 5 x^{2} - 3 x^{3} + 9 x^{4} )$
	$1 - 4 x + 11 x^{2} - 21 x^{3} + 33 x^{4} - 36 x^{5} + 27 x^{6}$
Frobenius angles:	$\pm0.166666666667$, $\pm0.345303779071$, $\pm0.557095674046$
Angle rank:	$2$ (numerical)
Jacobians:	$1$
Isomorphism classes:	3

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$11$	$1463$	$24332$	$551551$	$16264336$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$16$	$33$	$84$	$275$	$757$	$2156$	$6740$	$20229$	$58771$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=2x^8+2x^7+2x^6+2x^5+2x^3+2x^2+2$

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 $\times$ 2.3.ab_f 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 : \(\Q(\sqrt{-3}) \).
• 2.3.ab_f : 4.0.2725.1.

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

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

• 1.729.cc : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 2.729.abb_ef : 4.0.2725.1.

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 $\times$ 2.9.j_bl. The endomorphism algebra for each factor is:

  • 1.9.ad : \(\Q(\sqrt{-3}) \).
  • 2.9.j_bl : 4.0.2725.1.
• Endomorphism algebra over $\F_{3^{3}}$
  The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 2.27.f_ab. The endomorphism algebra for each factor is:

  • 1.27.a : \(\Q(\sqrt{-3}) \).
  • 2.27.f_ab : 4.0.2725.1.

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
3.3.ac_f_aj	$2$	3.9.g_t_bz
3.3.e_l_v	$2$	3.9.g_t_bz
3.3.ab_i_ag	$3$	(not in LMFDB)
3.3.c_f_j	$3$	(not in LMFDB)