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

• Label: 2.173.aby_bli
• 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:	$( 1 - 26 x + 173 x^{2} )( 1 - 24 x + 173 x^{2} )$
	$1 - 50 x + 970 x^{2} - 8650 x^{3} + 29929 x^{4}$
Frobenius angles:	$\pm0.0485897903475$, $\pm0.134271185755$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$22200$	$879120000$	$26780550708600$	$802321483392000000$	$24013782244908064851000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$124$	$29370$	$5172268$	$895702958$	$154963726844$	$26808755508810$	$4637914394666348$	$802359179605540318$	$138808137890442186364$	$24013807852744211285850$

Jacobians and polarizations

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

• $y^2=104x^6+29x^5+33x^4+153x^3+33x^2+29x+104$
• $y^2=143x^6+75x^5+55x^4+103x^3+55x^2+75x+143$
• $y^2=28x^6+82x^5+169x^4+117x^3+40x^2+69x+26$
• $y^2=12x^6+94x^5+40x^4+109x^3+14x^2+2x+98$
• $y^2=18x^6+149x^5+130x^4+52x^3+130x^2+149x+18$
• $y^2=154x^6+18x^5+30x^4+21x^3+30x^2+18x+154$
• $y^2=172x^6+99x^5+83x^4+125x^3+83x^2+99x+172$
• $y^2=143x^6+99x^5+53x^4+79x^3+20x^2+46x+53$
• $y^2=66x^6+128x^5+139x^4+117x^3+55x^2+53x+102$
• $y^2=35x^6+29x^5+31x^4+40x^3+122x^2+164x+40$
• $y^2=69x^6+47x^5+78x^4+132x^3+78x^2+47x+69$
• $y^2=171x^6+73x^5+10x^4+65x^3+124x^2+21x+127$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{173}$

The isogeny class factors as 1.173.aba $\times$ 1.173.ay 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.aba : \(\Q(\sqrt{-1}) \).
• 1.173.ay : \(\Q(\sqrt{-29}) \).

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.173.ac_aks	$2$	(not in LMFDB)
2.173.c_aks	$2$	(not in LMFDB)
2.173.by_bli	$2$	(not in LMFDB)