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

• Label: 2.121.abm_xd
• Base field: $\F_{11^{2}}$
• 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:	$\F_{11^{2}}$
Dimension:	$2$
L-polynomial:	$1 - 38 x + 601 x^{2} - 4598 x^{3} + 14641 x^{4}$
Frobenius angles:	$\pm0.121596203415$, $\pm0.205175270391$
Angle rank:	$2$ (numerical)
Number field:	4.0.752192.1
Galois group:	$D_{4}$
Jacobians:	$8$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10607$	$210856553$	$3138161605628$	$45954768981170393$	$672761101994682854607$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$14400$	$1771410$	$214382388$	$25937852824$	$3138433366278$	$379749876130968$	$45949730106476004$	$5559917313823774338$	$672749994921755783440$

Jacobians and polarizations

This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

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

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

The endomorphism algebra of this simple isogeny class is 4.0.752192.1.

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.bm_xd	$2$	(not in LMFDB)