Abelian variety isogeny class 2.1024.aex_izw over $\F_{2^{10}}$

• Label: 2.1024.aex_izw
• Base field: $\F_{2^{10}}$
• 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:	$\F_{2^{10}}$
Dimension:	$2$
L-polynomial:	$( 1 - 32 x )^{2}( 1 - 63 x + 1024 x^{2} )$
	$1 - 127 x + 6080 x^{2} - 130048 x^{3} + 1048576 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.0563432964760$
Angle rank:	$1$ (numerical)

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$924482$	$1095355857024$	$1152790463489319746$	$1208921762157280348320000$	$1267650476822616254571456465602$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$898$	$1044608$	$1073619778$	$1099507937536$	$1125899797236418$	$1152921501413353856$	$1180591620626310509122$	$1208925819612091242144256$	$1237940039285311473337833922$	$1267650600228227595472541145728$

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 $\F_{2^{10}}$.

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

The isogeny class factors as 1.1024.acm $\times$ 1.1024.acl and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.1024.acm : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.1024.acl : \(\Q(\sqrt{-127}) \).

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
2.1024.ab_acyi	$2$	(not in LMFDB)
2.1024.b_acyi	$2$	(not in LMFDB)
2.1024.ex_izw	$2$	(not in LMFDB)
2.1024.abf_bg	$3$	(not in LMFDB)