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

• Label: 2.13.b_p
• 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:	$1 + x + 15 x^{2} + 13 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.370469867856$, $\pm0.679487915055$
Angle rank:	$2$ (numerical)
Number field:	4.0.40725.3
Galois group:	$D_{4}$
Jacobians:	$16$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$199$	$34029$	$4812019$	$822446901$	$137597422864$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$15$	$199$	$2193$	$28795$	$370590$	$4819183$	$62764521$	$815798899$	$10604409279$	$137858640214$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):

• $y^2=6 x^6+5 x^5+4 x^4+2 x^3+7 x^2+7$
• $y^2=3 x^6+8 x^5+8 x^4+4 x^3+9 x^2+x+5$
• $y^2=4 x^6+10 x^5+3 x^4+8 x^2+3 x+5$
• $y^2=12 x^5+10 x^4+9 x^3+6 x^2+3 x+1$
• $y^2=12 x^6+8 x^5+3 x^4+10 x^3+6 x^2+3 x+12$
• $y^2=3 x^6+7 x^5+6 x^4+3 x^2+3 x$
• $y^2=4 x^6+5 x^5+10 x^4+x^2+6 x+3$
• $y^2=6 x^6+8 x^5+8 x^4+7 x^3+3 x^2+4 x+1$
• $y^2=12 x^6+5 x^5+9 x^4+9 x^3+6 x^2+4 x+6$
• $y^2=x^6+8 x^5+10 x^4+8 x^3+9 x+6$
• $y^2=2 x^6+6 x^5+x^4+5 x^3+10 x+2$
• $y^2=11 x^6+5 x^5+2 x^4+3 x^3+10 x+4$
• $y^2=3 x^6+6 x^5+11 x^4+x^3+3 x^2+2 x+8$
• $y^2=6 x^5+3 x^4+6 x^3+6 x^2+x+12$
• $y^2=7 x^6+5 x^5+11 x^3+x^2+6 x$
• $y^2=4 x^6+2 x^5+2 x^3+6 x^2+10 x+7$

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 4.0.40725.3.

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.ab_p	$2$	2.169.bd_ur