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

• Label: 2.11.b_aj
• 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:	$1 + x - 9 x^{2} + 11 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.221564005517$, $\pm0.870299311731$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\zeta_{5})\)
Galois group:	$C_4$
Jacobians:	$7$
Cyclic group of points:	no
Non-cyclic primes:	$5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$125$	$12625$	$1856375$	$217920125$	$26001250000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$13$	$103$	$1393$	$14883$	$161448$	$1774963$	$19477303$	$214365843$	$2357766583$	$25937365398$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which all are hyperelliptic):

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

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.ab_aj	$2$	2.121.at_lp
2.11.aj_bp	$5$	(not in LMFDB)
2.11.ae_g	$5$	(not in LMFDB)