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

• Label: 2.256.acl_cfw
• Base field: $\F_{2^{8}}$
• 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:	$\F_{2^{8}}$
Dimension:	$2$
L-polynomial:	$( 1 - 16 x )^{2}( 1 - 31 x + 256 x^{2} )$
	$1 - 63 x + 1504 x^{2} - 16128 x^{3} + 65536 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.0797861753495$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$50850$	$4232347200$	$281237242226850$	$18445878221841820800$	$1208922793868854295921250$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$194$	$64576$	$16763042$	$4294765696$	$1099508875874$	$281474940914368$	$72057593599175714$	$18446744068735211776$	$4722366482819171313122$	$1208925819614200475834176$

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 $\F_{2^{8}}$.

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

The isogeny class factors as 1.256.abg $\times$ 1.256.abf 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.abg : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• 1.256.abf : \(\Q(\sqrt{-7}) \).

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
2.256.ab_asm	$2$	(not in LMFDB)
2.256.b_asm	$2$	(not in LMFDB)
2.256.cl_cfw	$2$	(not in LMFDB)
2.256.ap_q	$3$	(not in LMFDB)