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

• Label: 2.11.ak_bv
• Base field: $\F_{11}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{11}$
Dimension:	$2$
L-polynomial:	$( 1 - 5 x + 11 x^{2} )^{2}$
	$1 - 10 x + 47 x^{2} - 110 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.228229222880$, $\pm0.228229222880$
Angle rank:	$1$ (numerical)
Jacobians:	$1$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$49$	$14161$	$1882384$	$221265625$	$26171797729$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$116$	$1412$	$15108$	$162502$	$1773686$	$19481842$	$214308868$	$2357756252$	$25937017556$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=2x^6+2x^5+x^3+2x+2$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{11}$

The isogeny class factors as 1.11.af^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.11.a_ad	$2$	2.121.ag_jr
2.11.k_bv	$2$	2.121.ag_jr
2.11.f_o	$3$	(not in LMFDB)