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

• Label: 2.256.ace_bxn
• 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:	$( 1 - 31 x + 256 x^{2} )( 1 - 25 x + 256 x^{2} )$
	$1 - 56 x + 1287 x^{2} - 14336 x^{3} + 65536 x^{4}$
Frobenius angles:	$\pm0.0797861753495$, $\pm0.214582404850$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$52432$	$4258317312$	$281434589337328$	$18446949261723340800$	$1208927345324213933319952$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$201$	$64975$	$16774809$	$4295015071$	$1099513015401$	$281474995242607$	$72057594132543801$	$18446744071919481151$	$4722366482822365359369$	$1208925819614427226704655$

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.az 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.az : \(\Q(\sqrt{-399}) \).

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.ag_akd	$2$	(not in LMFDB)
2.256.g_akd	$2$	(not in LMFDB)
2.256.ce_bxn	$2$	(not in LMFDB)