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

• Label: 2.16.ai_bt
• 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:	$1 - 8 x + 45 x^{2} - 128 x^{3} + 256 x^{4}$
Frobenius angles:	$\pm0.245740075077$, $\pm0.408506512405$
Angle rank:	$2$ (numerical)
Number field:	4.0.263952.2
Galois group:	$D_{4}$
Jacobians:	$16$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$166$	$72708$	$17547694$	$4314783552$	$1098998890006$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$283$	$4281$	$65839$	$1048089$	$16776619$	$268444185$	$4294924255$	$68718788025$	$1099509325243$

Jacobians and polarizations

This isogeny class contains the Jacobians of 16 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.263952.2.

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_bt	$2$	2.256.ba_sv