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

• Label: 2.11.aj_bp
• 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:	$1 - 9 x + 41 x^{2} - 99 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.178435994483$, $\pm0.329700688269$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\zeta_{5})\)
Galois group:	$C_4$
Jacobians:	$3$
Isomorphism classes:	3

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$55$	$14905$	$1882705$	$218522205$	$26001250000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$123$	$1413$	$14923$	$161448$	$1771503$	$19488423$	$214377763$	$2358012573$	$25937365398$

Jacobians and polarizations

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

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

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.j_bp	$2$	2.121.b_fl
2.11.ae_g	$5$	(not in LMFDB)
2.11.b_aj	$5$	(not in LMFDB)