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

• Label: 2.11.ag_v
• 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:	$1 - 6 x + 21 x^{2} - 66 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.120671085495$, $\pm0.507787998454$
Angle rank:	$2$ (numerical)
Number field:	4.0.424000.2
Galois group:	$D_{4}$
Jacobians:	$2$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$71$	$15265$	$1726436$	$210672265$	$25994784191$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$128$	$1296$	$14388$	$161406$	$1775918$	$19493466$	$214359268$	$2358062496$	$25937993648$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.424000.2.

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.g_v	$2$	2.121.g_aef