Abelian variety isogeny class 1.1024.acm over $\F_{2^{10}}$

• Label: 1.1024.acm
• Base field: $\F_{2^{10}}$
• Dimension: $1$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{10}}$
Dimension:	$1$
L-polynomial:	$( 1 - 32 x )^{2}$
	$1 - 64 x + 1024 x^{2}$
Frobenius angles:	$0$, $0$
Angle rank:	$0$ (numerical)
Number field:	\(\Q\)
Galois group:	Trivial

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$961$	$1046529$	$1073676289$	$1099509530625$	$1125899839733761$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$961$	$1046529$	$1073676289$	$1099509530625$	$1125899839733761$	$1152921502459363329$	$1180591620648691826689$	$1208925819612430151450625$	$1237940039285309906154946561$	$1267650600228227149696889520129$

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 endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

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
1.1024.cm	$2$	(not in LMFDB)
1.1024.bg	$3$	(not in LMFDB)
1.1024.a	$4$	(not in LMFDB)