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

• Label: 2.11.am_cg
• 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:	$( 1 - 6 x + 11 x^{2} )^{2}$
	$1 - 12 x + 58 x^{2} - 132 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.140218899004$, $\pm0.140218899004$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$36$	$11664$	$1726596$	$215737344$	$26090648676$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$94$	$1296$	$14734$	$162000$	$1776238$	$19504800$	$214413214$	$2358079776$	$25937619454$

Jacobians and polarizations

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

• $y^2=2x^6+x^4+x^2+2$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{11}$

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

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_ao	$2$	2.121.abc_qw
2.11.m_cg	$2$	2.121.abc_qw
2.11.g_z	$3$	(not in LMFDB)