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

• Label: 2.3.ab_e
• Base field: $\F_{3}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3}$
Dimension:	$2$
L-polynomial:	$( 1 - 2 x + 3 x^{2} )( 1 + x + 3 x^{2} )$
	$1 - x + 4 x^{2} - 3 x^{3} + 9 x^{4}$
Frobenius angles:	$\pm0.304086723985$, $\pm0.593214749339$
Angle rank:	$2$ (numerical)
Jacobians:	$1$
Isomorphism classes:	5

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:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10$	$180$	$760$	$7200$	$66550$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$17$	$30$	$89$	$273$	$674$	$2019$	$6641$	$20010$	$59057$

Jacobians and polarizations

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

• $y^2=2x^6+2x^4+x^3+2x^2+x$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{3}$

The isogeny class factors as 1.3.ac $\times$ 1.3.b 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 : \(\Q(\sqrt{-2}) \).
• 1.3.b : \(\Q(\sqrt{-11}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.3.ad_i	$2$	2.9.h_bc
2.3.b_e	$2$	2.9.h_bc
2.3.d_i	$2$	2.9.h_bc