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

• Label: 2.11.ai_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:	$\pm0.293962833700$, $\pm0.293962833700$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$64$	$16384$	$1960000$	$220463104$	$25962232384$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$134$	$1468$	$15054$	$161204$	$1767638$	$19469804$	$214332574$	$2358033508$	$25938057254$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{11}$

The isogeny class factors as 1.11.ae^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.a_g	$2$	2.121.m_ks
2.11.i_bm	$2$	2.121.m_ks
2.11.e_f	$3$	(not in LMFDB)