Abelian variety isogeny class 3.13.j_cj_jn over $\F_{13}$

• Label: 3.13.j_cj_jn
• Base field: $\F_{13}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{13}$
Dimension:	$3$
L-polynomial:	$1 + 9 x + 61 x^{2} + 247 x^{3} + 793 x^{4} + 1521 x^{5} + 2197 x^{6}$
Frobenius angles:	$\pm0.538567305886$, $\pm0.626855622023$, $\pm0.764770733179$
Angle rank:	$3$ (numerical)
Number field:	6.0.11945115039.1
Galois group:	$S_4\times C_2$
Cyclic group of points:	yes

This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$2$
Slopes:	$[0, 0, 1/2, 1/2, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$4829$	$6156975$	$9785046161$	$23331148710375$	$51291910870585209$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$23$	$211$	$2021$	$28603$	$372063$	$4828087$	$62738216$	$815702227$	$10604785523$	$137857926911$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

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

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 6.0.11945115039.1.

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
3.13.aj_cj_ajn	$2$	(not in LMFDB)