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

• Label: 2.32.ao_dt
• 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:	$( 1 - 11 x + 32 x^{2} )( 1 - 3 x + 32 x^{2} )$
	$1 - 14 x + 97 x^{2} - 448 x^{3} + 1024 x^{4}$
Frobenius angles:	$\pm0.0751336404065$, $\pm0.414573556089$
Angle rank:	$2$ (numerical)
Jacobians:	$40$
Isomorphism classes:	340

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$660$	$1045440$	$1073276820$	$1097231097600$	$1125373420449300$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$19$	$1023$	$32755$	$1046399$	$33538739$	$1073729151$	$34360121843$	$1099513425151$	$35184370447795$	$1125899894297343$

Jacobians and polarizations

This isogeny class contains the Jacobians of 40 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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

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.al $\times$ 1.32.ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.32.al : \(\Q(\sqrt{-7}) \).
• 1.32.ad : \(\Q(\sqrt{-119}) \).

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
2.32.ai_bf	$2$	2.1024.ac_abpv
2.32.i_bf	$2$	2.1024.ac_abpv
2.32.o_dt	$2$	2.1024.ac_abpv