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

• Label: 2.16.aj_bv
• 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:	$1 - 9 x + 47 x^{2} - 144 x^{3} + 256 x^{4}$
Frobenius angles:	$\pm0.177258786107$, $\pm0.410961538347$
Angle rank:	$2$ (numerical)
Number field:	4.0.466137.1
Galois group:	$D_{4}$
Jacobians:	$8$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$151$	$69007$	$17229100$	$4300861275$	$1099406044381$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$270$	$4205$	$65626$	$1048478$	$16784367$	$268489628$	$4295081746$	$68719015685$	$1099508069430$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.466137.1.

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_bv	$2$	2.256.n_ez