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

• Label: 3.3.ae_l_ay
• 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:	$( 1 + 3 x^{2} )( 1 - 4 x + 8 x^{2} - 12 x^{3} + 9 x^{4} )$
	$1 - 4 x + 11 x^{2} - 24 x^{3} + 33 x^{4} - 36 x^{5} + 27 x^{6}$
Frobenius angles:	$\pm0.0540867239847$, $\pm0.445913276015$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$1$
Isomorphism classes:	5

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$8$	$1088$	$17528$	$295936$	$12024808$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$16$	$24$	$36$	$200$	$784$	$2240$	$6332$	$19392$	$59536$

Jacobians and polarizations

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

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

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.a $\times$ 2.3.ae_i 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.a : \(\Q(\sqrt{-3}) \).
• 2.3.ae_i : \(\Q(\zeta_{8})\).

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

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

• 1.81.as : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
• 1.81.ao^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$

Remainder of endomorphism lattice by field

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

  • 1.9.g : the quaternion algebra over \(\Q\) ramified at $3$ and $\infty$.
  • 2.9.a_ao : \(\Q(\zeta_{8})\).

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.e_l_y	$2$	3.9.g_af_adg
3.3.ah_x_abw	$3$	(not in LMFDB)
3.3.ab_ab_a	$3$	(not in LMFDB)