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

• Label: 2.13.aj_br
• 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:	$1 - 9 x + 43 x^{2} - 117 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.161492811255$, $\pm0.377973052280$
Angle rank:	$2$ (numerical)
Number field:	4.0.92781.3
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$87$	$29493$	$5008851$	$819698949$	$137804618832$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$175$	$2279$	$28699$	$371150$	$4828255$	$62770055$	$815822419$	$10604591957$	$137857657750$

Jacobians and polarizations

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

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

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 4.0.92781.3.

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.j_br	$2$	2.169.f_dd