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

• Label: 2.11.ac_x
• 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:	$( 1 - x + 11 x^{2} )^{2}$
	$1 - 2 x + 23 x^{2} - 22 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.451829325548$, $\pm0.451829325548$
Angle rank:	$1$ (numerical)
Jacobians:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$121$	$20449$	$1860496$	$208600249$	$25760571001$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$164$	$1396$	$14244$	$159950$	$1774838$	$19502570$	$214338244$	$2357757676$	$25937461604$

Jacobians and polarizations

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

• $y^2=4x^6+10x^5+4x^4+4x^3+4x^2+10x+4$
• $y^2=7x^6+5x^5+9x^4+8x^3+5x^2+6x+7$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{11}$

The isogeny class factors as 1.11.ab^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$

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_v	$2$	2.121.bq_bah
2.11.c_x	$2$	2.121.bq_bah
2.11.b_ak	$3$	(not in LMFDB)