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

• Label: 2.32.a_a
• Base field: $\F_{2^{5}}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{5}}$
Dimension:	$2$
L-polynomial:	$( 1 - 8 x + 32 x^{2} )( 1 + 8 x + 32 x^{2} )$
	$1 + 1024 x^{4}$
Frobenius angles:	$\pm0.250000000000$, $\pm0.750000000000$
Angle rank:	$0$ (numerical)
Jacobians:	$31$

This isogeny class is not simple, not primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1025$	$1050625$	$1073741825$	$1103812890625$	$1125899906842625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$33$	$1025$	$32769$	$1052673$	$33554433$	$1073741825$	$34359738369$	$1099507433473$	$35184372088833$	$1125899906842625$

Jacobians and polarizations

This isogeny class contains the Jacobians of 31 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{20}}$.

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

The isogeny class factors as 1.32.ai $\times$ 1.32.i 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.ai : \(\Q(\sqrt{-1}) \).
• 1.32.i : \(\Q(\sqrt{-1}) \).

Endomorphism algebra over $\overline{\F}_{2^{5}}$

The base change of $A$ to $\F_{2^{20}}$ is 1.1048576.dau^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{10}}$

  The base change of $A$ to $\F_{2^{10}}$ is 1.1024.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{5}}$.

Subfield	Primitive Model
$\F_{2}$	2.2.a_a

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.32.aq_ey	$2$	2.1024.a_dau
2.32.q_ey	$2$	2.1024.a_dau
2.32.ai_bg	$6$	(not in LMFDB)