Abelian variety isogeny class 2.173.abt_bgo over $\F_{173}$

• Label: 2.173.abt_bgo
• Base field: $\F_{173}$
• 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:	$\F_{173}$
Dimension:	$2$
L-polynomial:	$( 1 - 25 x + 173 x^{2} )( 1 - 20 x + 173 x^{2} )$
	$1 - 45 x + 846 x^{2} - 7785 x^{3} + 29929 x^{4}$
Frobenius angles:	$\pm0.100717649571$, $\pm0.225058830207$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$22946$	$885853276$	$26807359396664$	$802394076574685056$	$24013921902477812598986$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$129$	$29597$	$5177448$	$895784001$	$154964628069$	$26808761356058$	$4637914376271753$	$802359178493292193$	$138808137875429458584$	$24013807852702050313397$

Jacobians and polarizations

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

• $y^2=75x^6+126x^5+154x^4+40x^3+93x^2+90x+28$
• $y^2=27x^6+7x^5+53x^4+30x^3+139x^2+53x+141$
• $y^2=144x^6+145x^4+78x^3+161x^2+82x+162$
• $y^2=69x^6+53x^5+52x^4+131x^3+83x^2+110x+79$
• $y^2=172x^6+154x^5+145x^4+69x^3+30x^2+83x+103$
• $y^2=17x^6+57x^5+114x^4+104x^3+88x^2+159x+69$
• $y^2=90x^6+18x^5+116x^4+130x^3+109x^2+103x+93$
• $y^2=136x^6+41x^5+69x^4+156x^3+69x^2+68x+81$
• $y^2=45x^6+24x^5+165x^4+28x^3+153x^2+158x+107$
• $y^2=133x^6+4x^5+96x^4+69x^3+103x^2+171x+86$
• $y^2=104x^6+33x^5+104x^4+98x^3+151x^2+88x+131$
• $y^2=76x^6+63x^5+72x^4+147x^3+103x^2+148x+24$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{173}$.

Endomorphism algebra over $\F_{173}$

The isogeny class factors as 1.173.az $\times$ 1.173.au and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.173.az : \(\Q(\sqrt{-67}) \).
• 1.173.au : \(\Q(\sqrt{-73}) \).

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
2.173.af_afy	$2$	(not in LMFDB)
2.173.f_afy	$2$	(not in LMFDB)
2.173.bt_bgo	$2$	(not in LMFDB)