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

• Label: 2.11.b_q
• 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:	$( 1 - 2 x + 11 x^{2} )( 1 + 3 x + 11 x^{2} )$
	$1 + x + 16 x^{2} + 11 x^{3} + 121 x^{4}$
Frobenius angles:	$\pm0.402508885479$, $\pm0.649384592723$
Angle rank:	$2$ (numerical)
Jacobians:	$10$
Isomorphism classes:	52
Cyclic group of points:	no
Non-cyclic primes:	$5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$150$	$18900$	$1751400$	$214250400$	$25900406250$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$13$	$153$	$1318$	$14633$	$160823$	$1768338$	$19495853$	$214405393$	$2357825458$	$25937097273$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

• $y^2=3 x^6+8 x^5+6 x^4+9 x^2+4 x+4$
• $y^2=7 x^6+2 x^5+9 x^4+10 x^3+10 x^2+9 x+7$
• $y^2=5 x^6+x^5+7 x^4+8 x^3+3 x^2+8 x+2$
• $y^2=4 x^6+5 x^5+8 x^4+2 x^3+10 x^2+8 x+3$
• $y^2=6 x^6+7 x^5+7 x^4+9 x^3+9 x^2+2 x$
• $y^2=6 x^6+2 x^5+2 x^4+x^2+x+8$
• $y^2=9 x^6+10 x^5+4 x^4+5 x^3+2 x^2+3 x+7$
• $y^2=5 x^6+8 x^5+7 x^4+6 x^3+6 x^2+8 x+1$
• $y^2=5 x^6+2 x^5+x^4+6 x^3+8 x^2+7 x+2$
• $y^2=4 x^6+x^5+4 x^4+9 x^3+4 x^2+5 x+5$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{11}$

The isogeny class factors as 1.11.ac $\times$ 1.11.d 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.ac : \(\Q(\sqrt{-10}) \).
• 1.11.d : \(\Q(\sqrt{-35}) \).

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.af_bc	$2$	2.121.bf_si
2.11.ab_q	$2$	2.121.bf_si
2.11.f_bc	$2$	2.121.bf_si