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

• Label: 2.8.ad_p
• 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:	$1 - 3 x + 15 x^{2} - 24 x^{3} + 64 x^{4}$
Frobenius angles:	$\pm0.301544139989$, $\pm0.517045260538$
Angle rank:	$2$ (numerical)
Number field:	4.0.113737.1
Galois group:	$D_{4}$
Jacobians:	$3$
Isomorphism classes:	3

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$53$	$5671$	$281324$	$16678411$	$1076657423$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$86$	$549$	$4074$	$32856$	$262271$	$2093370$	$16767538$	$134242893$	$1073863646$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+(a+1)x^5+(a+1)x^4+(a^2+a+1)x^3+(a^2+a+1)x^2+a+1$
• $y^2+(x^3+(a^2+a+1)x+a^2+a+1)y=x^6+(a^2+1)x^5+(a^2+1)x^4+(a+1)x^3+(a+1)x^2+a^2+1$
• $y^2+(x^3+(a+1)x+a+1)y=x^6+(a^2+a+1)x^5+(a^2+a+1)x^4+(a^2+1)x^3+(a^2+1)x^2+a^2+a+1$

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 4.0.113737.1.

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.d_p	$2$	2.64.v_ib