Abelian variety isogeny class 2.169.abx_bkc over $\F_{13^{2}}$

• Label: 2.169.abx_bkc
• Base field: $\F_{13^{2}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{13^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 25 x + 169 x^{2} )( 1 - 24 x + 169 x^{2} )$
	$1 - 49 x + 938 x^{2} - 8281 x^{3} + 28561 x^{4}$
Frobenius angles:	$\pm0.0885687144757$, $\pm0.125665916378$
Angle rank:	$2$ (numerical)
Jacobians:	$0$

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21170$	$800861100$	$23275867374440$	$665396405968320000$	$19004985654776977990850$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$121$	$28037$	$4822204$	$815705953$	$137858650561$	$23298092984882$	$3937376548525849$	$665416611808463233$	$112455406988440556476$	$19004963775329767350677$

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 $\F_{13^{2}}$.

Endomorphism algebra over $\F_{13^{2}}$

The isogeny class factors as 1.169.az $\times$ 1.169.ay and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.169.az : \(\Q(\sqrt{-51}) \).
• 1.169.ay : \(\Q(\sqrt{-1}) \).

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
2.169.ab_akc	$2$	(not in LMFDB)
2.169.b_akc	$2$	(not in LMFDB)
2.169.bx_bkc	$2$	(not in LMFDB)