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

• Label: 2.11.i_bm
• 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:	$( 1 + 4 x + 11 x^{2} )^{2}$
	$1 + 8 x + 38 x^{2} + 88 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.706037166300$, $\pm0.706037166300$
Angle rank:	$1$ (numerical)
Jacobians:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$256$	$16384$	$1597696$	$220463104$	$25913272576$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$20$	$134$	$1196$	$15054$	$160900$	$1767638$	$19504540$	$214332574$	$2357861876$	$25938057254$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{11}$

The isogeny class factors as 1.11.e^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$

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.ai_bm	$2$	2.121.m_ks
2.11.a_g	$2$	2.121.m_ks
2.11.ae_f	$3$	(not in LMFDB)