Abelian variety isogeny class 1.512.bg over $\F_{2^{9}}$

• Label: 1.512.bg
• Base field: $\F_{2^{9}}$
• Dimension: $1$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: yes
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{9}}$
Dimension:	$1$
L-polynomial:	$1 + 32 x + 512 x^{2}$
Frobenius angles:	$\pm0.750000000000$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\sqrt{-1}) \)
Galois group:	$C_2$

This isogeny class is simple and geometrically simple, not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$545$	$262145$	$134201345$	$68720001025$	$35184363700225$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$545$	$262145$	$134201345$	$68720001025$	$35184363700225$	$18014398509481985$	$9223372041149743105$	$4722366482732206260225$	$2417851639231457372667905$	$1237940039285380274899124225$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{36}}$.

Endomorphism algebra over $\F_{2^{9}}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1}) \).

Endomorphism algebra over $\overline{\F}_{2^{9}}$

The base change of $A$ to $\F_{2^{36}}$ is the simple isogeny class 1.68719476736.bdvoy and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{18}}$

  The base change of $A$ to $\F_{2^{18}}$ is the simple isogeny class 1.262144.a and its endomorphism algebra is \(\Q(\sqrt{-1}) \).

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{9}}$.

Subfield	Primitive Model
$\F_{2}$	1.2.c

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
1.512.abg	$2$	(not in LMFDB)
1.512.a	$8$	(not in LMFDB)