Abelian variety isogeny class 2.169.abu_bhb over $\F_{13^{2}}$

• Label: 2.169.abu_bhb
• Base field: $\F_{13^{2}}$
• 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:	$\F_{13^{2}}$
Dimension:	$2$
L-polynomial:	$1 - 46 x + 859 x^{2} - 7774 x^{3} + 28561 x^{4}$
Frobenius angles:	$\pm0.0365882703553$, $\pm0.217331560748$
Angle rank:	$2$ (numerical)
Number field:	4.0.153152.1
Galois group:	$D_{4}$
Jacobians:	$9$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21601$	$804442841$	$23287872895552$	$665417575590166313$	$19004976451350481472641$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$124$	$28164$	$4824694$	$815731908$	$137858583804$	$23298083251086$	$3937376290281628$	$665416607082473604$	$112455406920114147142$	$19004963774532188818564$

Jacobians and polarizations

This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=11ax^6+(10a+2)x^5+(11a+1)x^4+(11a+5)x^3+(4a+9)x^2+(3a+9)x+9a+4$
• $y^2=(3a+9)x^6+(a+2)x^5+(4a+12)x^4+(11a+10)x^3+(5a+4)x^2+(a+10)x+4a+1$
• $y^2=(5a+4)x^6+(10a+4)x^5+(8a+10)x^4+(12a+10)x^3+(4a+10)x^2+(3a+10)x+10a+9$
• $y^2=9ax^6+6ax^5+7ax^4+8ax^3+4ax+8a$
• $y^2=11ax^6+(6a+1)x^5+(7a+1)x^4+(8a+1)x^3+(2a+8)x^2+11ax+12a+5$
• $y^2=(8a+2)x^6+10x^5+(5a+9)x^4+(a+7)x^3+(3a+12)x^2+(5a+8)x+11a+5$
• $y^2=(8a+2)x^6+2x^5+(2a+3)x^4+(10a+1)x^3+(10a+7)x^2+10x+4a+7$
• $y^2=(9a+9)x^6+2x^5+(3a+1)x^4+(3a+5)x^3+(a+9)x^2+11a+6$
• $y^2=(9a+3)x^6+3x^5+(2a+8)x^4+(9a+10)x^3+(10a+5)x^2+(4a+9)x+12a+3$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13^{2}}$.

Endomorphism algebra over $\F_{13^{2}}$

The endomorphism algebra of this simple isogeny class is 4.0.153152.1.

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
2.169.bu_bhb	$2$	(not in LMFDB)