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

• Label: 2.16.aj_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:	$1 - 9 x + 49 x^{2} - 144 x^{3} + 256 x^{4}$
Frobenius angles:	$\pm0.211195784157$, $\pm0.390536017683$
Angle rank:	$2$ (numerical)
Number field:	4.0.2873.1
Galois group:	$D_{4}$
Jacobians:	$15$
Isomorphism classes:	15

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$153$	$70227$	$17454852$	$4318188003$	$1099500203673$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$274$	$4259$	$65890$	$1048568$	$16778815$	$268459640$	$4295014978$	$68718953315$	$1099507653874$

Jacobians and polarizations

This isogeny class contains the Jacobians of 15 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.2873.1.

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.j_bx	$2$	2.256.r_mj