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

• Label: 2.13.ad_r
• 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:	$1 - 3 x + 17 x^{2} - 39 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.264943912837$, $\pm0.582772800111$
Angle rank:	$2$ (numerical)
Number field:	4.0.34525.1
Galois group:	$D_{4}$
Jacobians:	$15$
Isomorphism classes:	15
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$145$	$33205$	$4846045$	$820329525$	$138532768000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$11$	$195$	$2207$	$28723$	$373106$	$4825635$	$62718947$	$815705443$	$10604574191$	$137858191350$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 15 curves (of which all are hyperelliptic):

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

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 4.0.34525.1.

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.d_r	$2$	2.169.z_pd