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

• Label: 2.11.al_bz
• Base field: $\F_{11}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{11}$
Dimension:	$2$
L-polynomial:	$1 - 11 x + 51 x^{2} - 121 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.0215640055172$, $\pm0.270299311731$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\zeta_{5})\)
Galois group:	$C_4$
Jacobians:	$1$
Isomorphism classes:	1

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$41$	$12505$	$1755251$	$214348205$	$25873696816$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$103$	$1321$	$14643$	$160656$	$1768123$	$19470991$	$214308243$	$2357847691$	$25937365398$

Jacobians and polarizations

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

• $y^2=8x^5+10$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{11}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{5})\).

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.l_bz	$2$	2.121.at_gz
2.11.ab_aj	$5$	(not in LMFDB)
2.11.ab_v	$5$	(not in LMFDB)