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

• Label: 2.16.af_y
• Base field: $\F_{2^{4}}$
• 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:	$\F_{2^{4}}$
Dimension:	$2$
L-polynomial:	$1 - 5 x + 24 x^{2} - 80 x^{3} + 256 x^{4}$
Frobenius angles:	$\pm0.212989141741$, $\pm0.550944557213$
Angle rank:	$2$ (numerical)
Number field:	4.0.77976.1
Galois group:	$D_{4}$
Jacobians:	$4$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$196$	$71736$	$16761724$	$4297990704$	$1103160477076$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$12$	$280$	$4092$	$65584$	$1052052$	$16787176$	$268406892$	$4294851424$	$68719490052$	$1099509640600$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.77976.1.

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.f_y	$2$	2.256.x_lc