Abelian variety isogeny class 2.9.ae_p over $\F_{3^{2}}$

• Label: 2.9.ae_p
• Base field: $\F_{3^{2}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{3^{2}}$
Dimension:	$2$
L-polynomial:	$1 - 4 x + 15 x^{2} - 36 x^{3} + 81 x^{4}$
Frobenius angles:	$\pm0.218106241758$, $\pm0.534324654653$
Angle rank:	$2$ (numerical)
Number field:	4.0.44688.2
Galois group:	$D_{4}$
Jacobians:	$2$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$57$	$7809$	$538308$	$43051017$	$3529298697$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$96$	$738$	$6564$	$59766$	$533430$	$4779606$	$43029060$	$387413874$	$3486740736$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=2ax^6+x^4+(2a+2)x^3+(a+1)x+2$
• $y^2=ax^6+2x^4+x^3+(2a+1)x+2a+2$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{3^{2}}$

The endomorphism algebra of this simple isogeny class is 4.0.44688.2.

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
2.9.e_p	$2$	2.81.o_dv