Abelian variety isogeny class 2.121.abn_xw over $\F_{11^{2}}$

• Label: 2.121.abn_xw
• Base field: $\F_{11^{2}}$
• 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:	$\F_{11^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 21 x + 121 x^{2} )( 1 - 18 x + 121 x^{2} )$
	$1 - 39 x + 620 x^{2} - 4719 x^{3} + 14641 x^{4}$
Frobenius angles:	$\pm0.0963413489042$, $\pm0.194982229042$
Angle rank:	$2$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10504$	$210290080$	$3136772587936$	$45952353597870720$	$672757843669116140104$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$83$	$14361$	$1770626$	$214371121$	$25937727203$	$3138432287118$	$379749869935883$	$45949730105662561$	$5559917314417897106$	$672749994932860926201$

Jacobians and polarizations

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

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

where $a$ is a root of the Conway polynomial.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{11^{2}}$

The isogeny class factors as 1.121.av $\times$ 1.121.as and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.121.av : \(\Q(\sqrt{-43}) \).
• 1.121.as : \(\Q(\sqrt{-10}) \).

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
2.121.ad_afg	$2$	(not in LMFDB)
2.121.d_afg	$2$	(not in LMFDB)
2.121.bn_xw	$2$	(not in LMFDB)