Abelian variety isogeny class 2.256.acg_bzx over $\F_{2^{8}}$

• Label: 2.256.acg_bzx
• Base field: $\F_{2^{8}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2^{8}}$
Dimension:	$2$
L-polynomial:	$( 1 - 31 x + 256 x^{2} )( 1 - 27 x + 256 x^{2} )$
	$1 - 58 x + 1349 x^{2} - 14848 x^{3} + 65536 x^{4}$
Frobenius angles:	$\pm0.0797861753495$, $\pm0.180343027596$
Angle rank:	$2$ (numerical)

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$51980$	$4251548160$	$281392292285180$	$18446801860866263040$	$1208927296503727709019500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$199$	$64871$	$16772287$	$4294980751$	$1099512970999$	$281475006914423$	$72057594500597359$	$18446744078861936671$	$4722366482904778617127$	$1208925819614606626236551$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{8}}$.

Endomorphism algebra over $\F_{2^{8}}$

The isogeny class factors as 1.256.abf $\times$ 1.256.abb and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.256.abf : \(\Q(\sqrt{-7}) \).
• 1.256.abb : \(\Q(\sqrt{-295}) \).

Base change

This is a primitive isogeny class.

Twists

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

Twist	Extension degree	Common base change
2.256.ae_amn	$2$	(not in LMFDB)
2.256.e_amn	$2$	(not in LMFDB)
2.256.cg_bzx	$2$	(not in LMFDB)