Abelian variety isogeny class 2.167.abt_bgc over $\F_{167}$

• Label: 2.167.abt_bgc
• Base field: $\F_{167}$
• 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:	$\F_{167}$
Dimension:	$2$
L-polynomial:	$( 1 - 25 x + 167 x^{2} )( 1 - 20 x + 167 x^{2} )$
	$1 - 45 x + 834 x^{2} - 7515 x^{3} + 27889 x^{4}$
Frobenius angles:	$\pm0.0816525061160$, $\pm0.218341865198$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21164$	$767914576$	$21686934588176$	$604984632225852736$	$16871991081777992266964$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$123$	$27533$	$4656384$	$777818841$	$129892471833$	$21691966535822$	$3622557602707719$	$604967116592714833$	$101029508526051139968$	$16871927924929070125253$

Jacobians and polarizations

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

• $y^2=43x^6+90x^5+138x^4+30x^3+67x^2+14x+7$
• $y^2=166x^6+147x^5+41x^4+104x^3+44x^2+61x+61$
• $y^2=82x^6+137x^5+132x^4+27x^3+52x^2+131x+22$
• $y^2=51x^6+79x^5+115x^4+15x^3+21x^2+15x+91$
• $y^2=4x^6+137x^5+133x^4+25x^3+149x^2+44x+32$
• $y^2=138x^6+138x^5+103x^4+40x^3+121x^2+83x+140$
• $y^2=166x^5+42x^4+25x^3+x^2+109x+83$
• $y^2=32x^6+82x^5+156x^4+146x^3+102x^2+57x+8$
• $y^2=140x^6+8x^5+39x^4+15x^3+62x^2+113x+36$
• $y^2=103x^6+146x^5+49x^4+159x^3+37x^2+139x+86$
• $y^2=162x^6+46x^5+73x^4+9x^3+13x^2+87x+128$
• $y^2=74x^6+124x^5+141x^4+110x^3+39x^2+3x+10$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{167}$.

Endomorphism algebra over $\F_{167}$

The isogeny class factors as 1.167.az $\times$ 1.167.au and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.167.az : \(\Q(\sqrt{-43}) \).
• 1.167.au : \(\Q(\sqrt{-67}) \).

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
2.167.af_agk	$2$	(not in LMFDB)
2.167.f_agk	$2$	(not in LMFDB)
2.167.bt_bgc	$2$	(not in LMFDB)