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

• Label: 2.8.ad_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:	$1 - 3 x + 13 x^{2} - 24 x^{3} + 64 x^{4}$
Frobenius angles:	$\pm0.266203455796$, $\pm0.544672082212$
Angle rank:	$2$ (numerical)
Number field:	4.0.243873.1
Galois group:	$D_{4}$
Jacobians:	$12$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$51$	$5355$	$271728$	$16841475$	$1087514871$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$82$	$531$	$4114$	$33186$	$262519$	$2092110$	$16766146$	$134232363$	$1073784682$

Jacobians and polarizations

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

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

where $a$ is a root of the Conway polynomial.

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 4.0.243873.1.

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_n	$2$	2.64.r_fx