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

• Label: 2.13.c_w
• 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:	$1 + 2 x + 22 x^{2} + 26 x^{3} + 169 x^{4}$
Frobenius angles:	$\pm0.445167147195$, $\pm0.648134864011$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-46 +2 \sqrt{5}})\)
Galois group:	$D_{4}$
Jacobians:	$18$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$220$	$36080$	$4725820$	$811078400$	$137844305500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$210$	$2152$	$28398$	$371256$	$4824930$	$62765152$	$815767518$	$10604115136$	$137858336050$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

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

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 \(\Q(\sqrt{-46 +2 \sqrt{5}})\).

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.ac_w	$2$	2.169.bo_bbq