Abelian variety isogeny class 2.8.ae_n over $\F_{2^{3}}$

• Label: 2.8.ae_n
• Base field: $\F_{2^{3}}$
• 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:	$\F_{2^{3}}$
Dimension:	$2$
L-polynomial:	$1 - 4 x + 13 x^{2} - 32 x^{3} + 64 x^{4}$
Frobenius angles:	$\pm0.193271217563$, $\pm0.536415660845$
Angle rank:	$2$ (numerical)
Number field:	4.0.257936.1
Galois group:	$D_{4}$
Jacobians:	$12$
Isomorphism classes:	12

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$42$	$4788$	$260946$	$16700544$	$1092097482$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$75$	$509$	$4079$	$33325$	$263835$	$2096365$	$16771039$	$134221757$	$1073704875$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

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

The endomorphism algebra of this simple isogeny class is 4.0.257936.1.

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
2.8.e_n	$2$	2.64.k_bp