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

• Label: 2.121.abm_xe
• 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:	$( 1 - 20 x + 121 x^{2} )( 1 - 18 x + 121 x^{2} )$
	$1 - 38 x + 602 x^{2} - 4598 x^{3} + 14641 x^{4}$
Frobenius angles:	$\pm0.136777651826$, $\pm0.194982229042$
Angle rank:	$2$ (numerical)
Jacobians:	$32$
Isomorphism classes:	96

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$10608$	$210887040$	$3138364081008$	$45955491499008000$	$672762885992818743408$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$84$	$14402$	$1771524$	$214385758$	$25937921604$	$3138434422562$	$379749888262164$	$45949730193499198$	$5559917313634913844$	$672749994902262403202$

Jacobians and polarizations

This isogeny class contains the Jacobians of 32 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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

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.au $\times$ 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.au : \(\Q(\sqrt{-21}) \).
• 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.ac_aeo	$2$	(not in LMFDB)
2.121.c_aeo	$2$	(not in LMFDB)
2.121.bm_xe	$2$	(not in LMFDB)