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

• Label: 2.8.c_f
• 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:	$1 + 2 x + 5 x^{2} + 16 x^{3} + 64 x^{4}$
Frobenius angles:	$\pm0.356537734320$, $\pm0.789478152311$
Angle rank:	$2$ (numerical)
Number field:	4.0.45072.1
Galois group:	$D_{4}$
Jacobians:	$18$
Isomorphism classes:	30

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$88$	$4576$	$275704$	$17370496$	$1053001048$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$11$	$71$	$539$	$4239$	$32131$	$261911$	$2096875$	$16780063$	$134258003$	$1073666471$

Jacobians and polarizations

This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.45072.1.

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.ac_f	$2$	2.64.g_dl