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

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

Invariants

Base field:	$\F_{2^{5}}$
Dimension:	$2$
L-polynomial:	$( 1 - 9 x + 32 x^{2} )( 1 - 8 x + 32 x^{2} )$
	$1 - 17 x + 136 x^{2} - 544 x^{3} + 1024 x^{4}$
Frobenius angles:	$\pm0.207210850837$, $\pm0.250000000000$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$600$	$1033200$	$1086654600$	$1103509260000$	$1126561191243000$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$1008$	$33160$	$1052384$	$33574136$	$1073789136$	$34359533672$	$1099508533696$	$35184352889560$	$1125899841448368$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

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.aj $\times$ 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.aj : \(\Q(\sqrt{-47}) \).
• 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.cpr $\times$ 1.1048576.dau. The endomorphism algebra for each factor is:

• 1.1048576.cpr : \(\Q(\sqrt{-47}) \).
• 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.ar $\times$ 1.1024.a. The endomorphism algebra for each factor is:

  • 1.1024.ar : \(\Q(\sqrt{-47}) \).
  • 1.1024.a : \(\Q(\sqrt{-1}) \).

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.ab_ai	$2$	2.1024.ar_dau
2.32.b_ai	$2$	2.1024.ar_dau
2.32.r_fg	$2$	2.1024.ar_dau