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

• Label: 2.13.ah_bi
• 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:	$1 - 7 x + 34 x^{2} - 91 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.219632524221$, $\pm0.436075769158$
Angle rank:	$2$ (numerical)
Number field:	4.0.304028.1
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$106$	$32012$	$5048992$	$817970624$	$137907422626$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$7$	$189$	$2296$	$28641$	$371427$	$4830762$	$62762287$	$815691489$	$10604093752$	$137857571509$

Jacobians and polarizations

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

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

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 4.0.304028.1.

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.h_bi	$2$	2.169.t_im