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

• Label: 2.11.ah_bb
• 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:	$1 - 7 x + 27 x^{2} - 77 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.116678659763$, $\pm0.461158112795$
Angle rank:	$2$ (numerical)
Number field:	4.0.206045.1
Galois group:	$D_{4}$
Jacobians:	$4$
Isomorphism classes:	4

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$65$	$15145$	$1764035$	$210894125$	$25897066000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$127$	$1325$	$14403$	$160800$	$1775107$	$19501235$	$214371123$	$2357957135$	$25937814102$

Jacobians and polarizations

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

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

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 4.0.206045.1.

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
2.11.h_bb	$2$	2.121.f_aed