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

• Label: 2.169.abt_bga
• Base field: $\F_{13^{2}}$
• 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:	$\F_{13^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 13 x )^{2}( 1 - 19 x + 169 x^{2} )$
	$1 - 45 x + 832 x^{2} - 7605 x^{3} + 28561 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.239161554446$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21744$	$805484736$	$23290263154944$	$665416176060384000$	$19004945008718963893104$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$125$	$28201$	$4825190$	$815730193$	$137858355725$	$23298077427406$	$3937376194922645$	$665416605980412193$	$112455406911925830710$	$19004963774513004494281$

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.aba $\times$ 1.169.at 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.aba : the quaternion algebra over \(\Q\) ramified at $13$ and $\infty$.
• 1.169.at : \(\Q(\sqrt{-35}) \).

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
2.169.ah_aga	$2$	(not in LMFDB)
2.169.h_aga	$2$	(not in LMFDB)
2.169.bt_bga	$2$	(not in LMFDB)