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

• Label: 2.11.ak_bu
• 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:	$( 1 - 6 x + 11 x^{2} )( 1 - 4 x + 11 x^{2} )$
	$1 - 10 x + 46 x^{2} - 110 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.140218899004$, $\pm0.293962833700$
Angle rank:	$2$ (numerical)
Jacobians:	$2$
Isomorphism classes:	8
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$48$	$13824$	$1839600$	$218087424$	$26026361328$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$114$	$1382$	$14894$	$161602$	$1771938$	$19487302$	$214372894$	$2358056642$	$25937838354$

Jacobians and polarizations

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

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

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 $\times$ 1.11.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.11.ag : \(\Q(\sqrt{-2}) \).
• 1.11.ae : \(\Q(\sqrt{-7}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.11.ac_ac	$2$	2.121.ai_gc
2.11.c_ac	$2$	2.121.ai_gc
2.11.k_bu	$2$	2.121.ai_gc