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

• Label: 2.1024.aep_ige
• 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: yes

Invariants

Base field:	$\F_{2^{10}}$
Dimension:	$2$
L-polynomial:	$( 1 - 32 x )^{2}( 1 - 55 x + 1024 x^{2} )$
	$1 - 119 x + 5568 x^{2} - 121856 x^{3} + 1048576 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.170852887823$
Angle rank:	$1$ (numerical)

This isogeny class is not simple, 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$932170$	$1096343780400$	$1152853913463294490$	$1208924770098697393500000$	$1267650592446413907299277375850$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$906$	$1045552$	$1073678874$	$1099510673248$	$1125899899930986$	$1152921504600164752$	$1180591620704793946554$	$1208925819613323587386048$	$1237940039285301596560653066$	$1267650600228225777790805504752$

Jacobians and polarizations

This isogeny class contains a Jacobian, and hence is principally polarizable.

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.acd 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.acd : \(\Q(\sqrt{-119}) \).

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.aj_aceq	$2$	(not in LMFDB)
2.1024.j_aceq	$2$	(not in LMFDB)
2.1024.ep_ige	$2$	(not in LMFDB)
2.1024.ax_lc	$3$	(not in LMFDB)