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

• Label: 2.32.at_fw
• Base field: $\F_{2^{5}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{5}}$
Dimension:	$2$
L-polynomial:	$( 1 - 11 x + 32 x^{2} )( 1 - 8 x + 32 x^{2} )$
	$1 - 19 x + 152 x^{2} - 608 x^{3} + 1024 x^{4}$
Frobenius angles:	$\pm0.0751336404065$, $\pm0.250000000000$
Angle rank:	$1$ (numerical)
Jacobians:	$1$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$550$	$992200$	$1073114350$	$1100399410000$	$1126026807127750$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$968$	$32750$	$1049424$	$33558214$	$1073731736$	$34359506398$	$1099510185376$	$35184369936950$	$1125899954494568$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2+xy=x^5+x^3+x^2+x$

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.al $\times$ 1.32.ai 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.ai : \(\Q(\sqrt{-1}) \).

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

The base change of $A$ to $\F_{2^{20}}$ is 1.1048576.abuf $\times$ 1.1048576.dau. The endomorphism algebra for each factor is:

• 1.1048576.abuf : \(\Q(\sqrt{-7}) \).
• 1.1048576.dau : 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.acf $\times$ 1.1024.a. The endomorphism algebra for each factor is:

  • 1.1024.acf : \(\Q(\sqrt{-7}) \).
  • 1.1024.a : \(\Q(\sqrt{-1}) \).

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.b_c

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.32.ad_ay	$2$	2.1024.acf_dau
2.32.d_ay	$2$	2.1024.acf_dau
2.32.t_fw	$2$	2.1024.acf_dau