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

• Label: 2.13.ak_bw
• Base field: $\F_{13}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{13}$
Dimension:	$2$
L-polynomial:	$1 - 10 x + 48 x^{2} - 130 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.116678169037$, $\pm0.350288405554$
Angle rank:	$2$ (numerical)
Number field:	4.0.39744.5
Galois group:	$D_{4}$
Jacobians:	$4$
Isomorphism classes:	4

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$78$	$27924$	$4937166$	$817726416$	$137654749518$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$166$	$2248$	$28630$	$370744$	$4825222$	$62759428$	$815833054$	$10604885524$	$137859128086$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=5x^6+10x^5+6x^4+11x^3+12x^2+9x+12$
• $y^2=11x^6+4x^5+4x^4+6x^3+7x^2+3x+11$
• $y^2=6x^6+4x^5+2x^4+8x^3+9x^2+2x+2$
• $y^2=x^6+9x^5+2x^3+3x^2+4x+9$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{13}$

The endomorphism algebra of this simple isogeny class is 4.0.39744.5.

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
2.13.k_bw	$2$	2.169.ae_bq