Abelian variety isogeny class 2.16.ak_bx over $\F_{2^{4}}$

• Label: 2.16.ak_bx
• Base field: $\F_{2^{4}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{4}}$
Dimension:	$2$
L-polynomial:	$1 - 10 x + 49 x^{2} - 160 x^{3} + 256 x^{4}$
Frobenius angles:	$\pm0.0660425289118$, $\pm0.412497962872$
Angle rank:	$2$ (numerical)
Number field:	4.0.10304.1
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	20

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$136$	$64736$	$16734664$	$4257298304$	$1096317656776$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$7$	$255$	$4087$	$64959$	$1045527$	$16773951$	$268462327$	$4295055999$	$68719305367$	$1099510702975$

Jacobians and polarizations

This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2+(x^2+x+a^3+a+1)y=ax^5+(a^3+1)x^4+(a^3+a^2)x^3+(a^3+1)x^2+(a^2+a+1)x+a^2+1$
• $y^2+(x^2+x+a^3+a^2+1)y=(a+1)x^5+(a^3+1)x^4+(a^3+a)x^3+(a^3+1)x^2+(a^2+a+1)x+a^3$
• $y^2+(x^2+x+a^3+a^2+a)y=(a^2+1)x^5+a^3x^4+a^3x^3+a^3x^2+(a^2+a)x+a^3+a^2+1$
• $y^2+(x^2+x+a^3+1)y=a^2x^5+a^3x^4+(a^3+a^2+a+1)x^3+a^3x^2+(a^2+a)x+a^3+a^2+a$
• $y^2+(x^2+x+a^3)y=ax^5+(a^3+a^2+a)x^4+(a^3+1)x^3+(a^3+a^2+a)x^2+(a^2+a+1)x+a+1$
• $y^2+(x^2+x)y=a^3x^5+(a^3+a^2+a)x^4+x^3+x^2+(a^2+a)x$
• $y^2+(x^2+x)y=(a^2+a+1)x^5+(a^3+1)x^2+(a^3+a^2+a)x$
• $y^2+(x^2+x+a^3+a^2)y=a^2x^5+(a^3+a^2+a+1)x^4+(a^3+a^2+1)x^3+(a^3+a^2+a+1)x^2+(a^2+a)x+a^3+a^2$
• $y^2+(x^2+x+a^3+a)y=(a^2+1)x^5+(a^3+a)x^4+(a^3+a+1)x^3+(a^3+a)x^2+(a^2+a)x+a^3+1$
• $y^2+(x^2+x)y=(a^3+a^2)x^5+x^3+(a^2+a+1)x^2+(a^3+a)x$
• $y^2+(x^2+x)y=(a^2+a)x^5+(a^3+a^2+1)x^2+(a^3+a+1)x$
• $y^2+(x^2+x+a^3+a^2+a+1)y=(a+1)x^5+(a^3+a^2+1)x^4+(a^3+a^2+a)x^3+(a^3+a^2+1)x^2+(a^2+a+1)x+a^3+a^2$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{4}}$.

Endomorphism algebra over $\F_{2^{4}}$

The endomorphism algebra of this simple isogeny class is 4.0.10304.1.

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
2.16.k_bx	$2$	2.256.ac_alb