Abelian variety isogeny class 2.3.ab_g over $\F_{3}$

• Label: 2.3.ab_g
• Base field: $\F_{3}$
• Dimension: $2$
• $p$-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:	$2$
L-polynomial:	$( 1 - x + 3 x^{2} )( 1 + 3 x^{2} )$
	$1 - x + 6 x^{2} - 3 x^{3} + 9 x^{4}$
Frobenius angles:	$\pm0.406785250661$, $\pm0.5$
Angle rank:	$1$ (numerical)
Jacobians:	$0$
Isomorphism classes:	2

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$12$	$240$	$1008$	$4800$	$51972$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$21$	$36$	$57$	$213$	$774$	$2271$	$6513$	$19548$	$59061$

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 $\F_{3^{2}}$.

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ab $\times$ 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.ab : \(\Q(\sqrt{-11}) \).
• 1.3.a : \(\Q(\sqrt{-3}) \).

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

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

• 1.9.f : \(\Q(\sqrt{-11}) \).
• 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
2.3.b_g	$2$	2.9.l_bw
2.3.ae_j	$3$	2.27.i_cc
2.3.c_d	$3$	2.27.i_cc