Abelian variety isogeny class 2.11.e_u over $\F_{11}$

• Label: 2.11.e_u
• Base field: $\F_{11}$
• 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:	$\F_{11}$
Dimension:	$2$
L-polynomial:	$1 + 4 x + 20 x^{2} + 44 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.478413784700$, $\pm0.734043019883$
Angle rank:	$2$ (numerical)
Number field:	4.0.610560.3
Galois group:	$D_{4}$
Jacobians:	$4$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$190$	$17860$	$1714750$	$214391440$	$25828984750$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$146$	$1288$	$14646$	$160376$	$1773218$	$19499216$	$214306846$	$2357923888$	$25937830226$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11}$.

Endomorphism algebra over $\F_{11}$

The endomorphism algebra of this simple isogeny class is 4.0.610560.3.

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
2.11.ae_u	$2$	2.121.y_le