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

• Label: 3.3.ae_j_aq
• Base field: $\F_{3}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• 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:	$( 1 - 2 x + 3 x^{2} )( 1 - 2 x + 2 x^{2} - 6 x^{3} + 9 x^{4} )$
	$1 - 4 x + 9 x^{2} - 16 x^{3} + 27 x^{4} - 36 x^{5} + 27 x^{6}$
Frobenius angles:	$\pm0.116139763599$, $\pm0.304086723985$, $\pm0.616139763599$
Angle rank:	$2$ (numerical)
Jacobians:	$4$
Isomorphism classes:	22

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8$	$960$	$16568$	$614400$	$16708648$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$12$	$24$	$92$	$280$	$684$	$2128$	$6844$	$20112$	$59532$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which 3 are hyperelliptic), and hence is principally polarizable:

• $y^2=2x^8+x^7+2x^6+2x^5+2x^2+2$
• $y^2=2x^8+2x^6+2x^4+2$
• $y^2=2x^8+x^7+2x^3+x^2+2$
• $x^4+x^2y^2+2x^2yz+2x^2z^2+2y^4+y^3z+z^4=0$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{4}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ac $\times$ 2.3.ac_c 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.ac : \(\Q(\sqrt{-2}) \).
• 2.3.ac_c : \(\Q(i, \sqrt{5})\).

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

The base change of $A$ to $\F_{3^{4}}$ is 1.81.ac^2 $\times$ 1.81.o. The endomorphism algebra for each factor is:

• 1.81.ac^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$
• 1.81.o : \(\Q(\sqrt{-2}) \).

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{3^{2}}$
  The base change of $A$ to $\F_{3^{2}}$ is 1.9.c $\times$ 2.9.a_ac. The endomorphism algebra for each factor is:

  • 1.9.c : \(\Q(\sqrt{-2}) \).
  • 2.9.a_ac : \(\Q(i, \sqrt{5})\).

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.a_b_ai	$2$	3.9.c_h_ae
3.3.a_b_i	$2$	3.9.c_h_ae
3.3.e_j_q	$2$	3.9.c_h_ae