Abelian variety isogeny class 2.32.as_fp over $\F_{2^{5}}$

• Label: 2.32.as_fp
• Base field: $\F_{2^{5}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{5}}$
Dimension:	$2$
L-polynomial:	$( 1 - 9 x + 32 x^{2} )^{2}$
	$1 - 18 x + 145 x^{2} - 576 x^{3} + 1024 x^{4}$
Frobenius angles:	$\pm0.207210850837$, $\pm0.207210850837$
Angle rank:	$1$ (numerical)
Jacobians:	$10$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$576$	$1016064$	$1082673216$	$1103205712896$	$1126672596611136$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$15$	$991$	$33039$	$1052095$	$33577455$	$1073836447$	$34359853263$	$1099509633919$	$35184350467503$	$1125899776054111$

Jacobians and polarizations

This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

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

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

The isogeny class factors as 1.32.aj^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-47}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.32.a_ar	$2$	2.1024.abi_dlx
2.32.s_fp	$2$	2.1024.abi_dlx
2.32.j_bx	$3$	(not in LMFDB)