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

• Label: 2.16.ai_bh
• 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:	$1 - 8 x + 33 x^{2} - 128 x^{3} + 256 x^{4}$
Frobenius angles:	$\pm0.0567971777196$, $\pm0.494945955570$
Angle rank:	$2$ (numerical)
Number field:	4.0.464400.1
Galois group:	$D_{4}$
Jacobians:	$8$
Isomorphism classes:	16

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$154$	$65604$	$16358650$	$4236443904$	$1097995594234$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$259$	$3993$	$64639$	$1047129$	$16781443$	$268428729$	$4294818559$	$68719420473$	$1099514142979$

Jacobians and polarizations

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

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

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 4.0.464400.1.

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.i_bh	$2$	2.256.c_arf