Abelian variety isogeny class 2.13.al_cc over $\F_{13}$

• Label: 2.13.al_cc
• Base field: $\F_{13}$
• 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:	$\F_{13}$
Dimension:	$2$
L-polynomial:	$( 1 - 7 x + 13 x^{2} )( 1 - 4 x + 13 x^{2} )$
	$1 - 11 x + 54 x^{2} - 143 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.0772104791556$, $\pm0.312832958189$
Angle rank:	$2$ (numerical)
Jacobians:	$1$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$70$	$26460$	$4873120$	$817084800$	$137610200350$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$157$	$2220$	$28609$	$370623$	$4822234$	$62737419$	$815751841$	$10604790060$	$137859735157$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):

• $y^2=5 x^6+2 x^5+3 x^4+11 x^3+5 x^2+3 x+5$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13}$.

Endomorphism algebra over $\F_{13}$

The isogeny class factors as 1.13.ah $\times$ 1.13.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.13.ah : \(\Q(\sqrt{-3}) \).
• 1.13.ae : \(\Q(\sqrt{-1}) \).

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
2.13.ad_ac	$2$	2.169.an_ee
2.13.d_ac	$2$	2.169.an_ee
2.13.l_cc	$2$	2.169.an_ee
2.13.ac_s	$3$	(not in LMFDB)
2.13.b_g	$3$	(not in LMFDB)