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

• Label: 2.121.abo_yo
• Base field: $\F_{11^{2}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• 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:	$( 1 - 11 x )^{2}( 1 - 18 x + 121 x^{2} )$
	$1 - 40 x + 638 x^{2} - 4840 x^{3} + 14641 x^{4}$
Frobenius angles:	$0$, $0$, $\pm0.194982229042$
Angle rank:	$1$ (numerical)
Jacobians:	$10$

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10400$	$209664000$	$3134957789600$	$45948288282624000$	$672749968943582660000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$82$	$14318$	$1769602$	$214352158$	$25937423602$	$3138427883918$	$379749810661282$	$45949729354708798$	$5559917305391377042$	$672749994829453514798$

Jacobians and polarizations

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

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

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.aw $\times$ 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.aw : the quaternion algebra over \(\Q\) ramified at $11$ and $\infty$.
• 1.121.as : \(\Q(\sqrt{-10}) \).

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
2.121.ae_afy	$2$	(not in LMFDB)
2.121.e_afy	$2$	(not in LMFDB)
2.121.bo_yo	$2$	(not in LMFDB)
2.121.ah_bs	$3$	(not in LMFDB)