Abelian variety isogeny class 1.32.b over $\F_{2^{5}}$

• Label: 1.32.b
• Base field: $\F_{2^{5}}$
• Dimension: $1$
• $p$-rank: $1$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{5}}$
Dimension:	$1$
L-polynomial:	$1 + x + 32 x^{2}$
Frobenius angles:	$\pm0.528171648238$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-127}) \)
Galois group:	$C_2$
Jacobians:	$5$
Isomorphism classes:	5
Cyclic group of points:	yes

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:	$1$
Slopes:	$[0, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$34$	$1088$	$32674$	$1046656$	$33559394$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$34$	$1088$	$32674$	$1046656$	$33559394$	$1073798336$	$34359523106$	$1099510034688$	$35184380570338$	$1125899949339968$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which 0 are hyperelliptic):

• $y^2+x y=x^3+a^{10} x^2+a^4$
• $y^2+x y=x^3+a^{10} x^2+a^{16}$
• $y^2+x y=x^3+x^2+a^8$
• $y^2+x y=x^3+a^{22} x^2+a^2$
• $y^2+x y=x^3+a^{22} x^2+a$

where $a$ is a root of the Conway polynomial.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{5}}$.

Endomorphism algebra over $\F_{2^{5}}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-127}) \).

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
1.32.ab	$2$	1.1024.cl