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

• Label: 2.11.a_s
• 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:	$( 1 - 2 x + 11 x^{2} )( 1 + 2 x + 11 x^{2} )$
	$1 + 18 x^{2} + 121 x^{4}$
Frobenius angles:	$\pm0.402508885479$, $\pm0.597491114521$
Angle rank:	$1$ (numerical)
Jacobians:	$12$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$140$	$19600$	$1770860$	$211993600$	$25937103500$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$12$	$158$	$1332$	$14478$	$161052$	$1770158$	$19487172$	$214403998$	$2357947692$	$25936782398$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{11^{2}}$.

Endomorphism algebra over $\F_{11}$

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

Endomorphism algebra over $\overline{\F}_{11}$

The base change of $A$ to $\F_{11^{2}}$ is 1.121.s^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$

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.ae_ba	$2$	2.121.bk_vu
2.11.e_ba	$2$	2.121.bk_vu
2.11.a_as	$4$	(not in LMFDB)