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

• Label: 3.3.ae_n_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:	$( 1 + 3 x^{2} )( 1 - 2 x + 3 x^{2} )^{2}$
	$1 - 4 x + 13 x^{2} - 24 x^{3} + 39 x^{4} - 36 x^{5} + 27 x^{6}$
Frobenius angles:	$\pm0.304086723985$, $\pm0.304086723985$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$1$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$16$	$2304$	$40432$	$589824$	$14289616$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$20$	$48$	$92$	$240$	$692$	$2016$	$6332$	$20064$	$60500$

Jacobians and polarizations

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

• $y^2=2x^8+x^7+x^5+x^4+2x^3+2x+2$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{3^{2}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ac^2 $\times$ 1.3.a 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^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
• 1.3.a : \(\Q(\sqrt{-3}) \).

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

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

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

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_f_a	$2$	3.9.k_cd_hw
3.3.e_n_y	$2$	3.9.k_cd_hw
3.3.ah_z_acc	$3$	(not in LMFDB)
3.3.ab_ac_j	$3$	(not in LMFDB)
3.3.ab_b_g	$3$	(not in LMFDB)
3.3.c_e_m	$3$	(not in LMFDB)
3.3.f_k_p	$3$	(not in LMFDB)