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

• Label: 2.173.abt_bgi
• 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:	$( 1 - 26 x + 173 x^{2} )( 1 - 19 x + 173 x^{2} )$
	$1 - 45 x + 840 x^{2} - 7785 x^{3} + 29929 x^{4}$
Frobenius angles:	$\pm0.0485897903475$, $\pm0.243098056104$
Angle rank:	$2$ (numerical)
Jacobians:	$18$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$22940$	$885484000$	$26803159497920$	$802368664178800000$	$24013814791080017410700$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$129$	$29585$	$5176638$	$895755633$	$154963936869$	$26808748368230$	$4637914180240953$	$802359176097859393$	$138808137852176460534$	$24013807852530545082425$

Jacobians and polarizations

This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=30x^6+32x^5+154x^4+75x^3+146x^2+39x+16$
• $y^2=87x^6+99x^5+91x^4+8x^3+88x^2+171x+1$
• $y^2=155x^6+89x^5+96x^4+145x^3+151x^2+138x+28$
• $y^2=38x^6+107x^5+34x^4+71x^3+96x^2+28x+110$
• $y^2=167x^6+37x^5+160x^4+135x^3+156x^2+41x+71$
• $y^2=49x^6+2x^5+108x^4+73x^3+35x^2+61x+69$
• $y^2=54x^6+165x^5+83x^4+43x^3+42x^2+171$
• $y^2=98x^6+9x^5+82x^4+79x^3+69x^2+133x+108$
• $y^2=9x^6+133x^5+114x^4+24x^3+38x^2+143x+111$
• $y^2=70x^6+28x^5+95x^4+148x^3+25x^2+57x+21$
• $y^2=158x^6+119x^5+58x^4+2x^3+166x^2+121x+103$
• $y^2=114x^6+34x^5+139x^4+36x^3+40x^2+52x+148$
• $y^2=8x^6+26x^5+95x^4+30x^3+157x^2+115x+40$
• $y^2=11x^6+107x^4+124x^3+133x^2+37x+117$
• $y^2=166x^6+42x^5+115x^4+155x^3+17x^2+105x+165$
• $y^2=117x^6+54x^5+51x^4+116x^3+23x^2+161x+92$
• $y^2=12x^6+106x^5+56x^4+82x^3+5x^2+157x+49$
• $y^2=138x^6+159x^5+81x^4+88x^3+93x^2+95x+13$

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.at 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.at : \(\Q(\sqrt{-331}) \).

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.ah_afs	$2$	(not in LMFDB)
2.173.h_afs	$2$	(not in LMFDB)
2.173.bt_bgi	$2$	(not in LMFDB)