# Properties

 Label 2.173.abt_bgg Base Field $\F_{173}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{173}$ Dimension: $2$ L-polynomial: $1 - 45 x + 838 x^{2} - 7785 x^{3} + 29929 x^{4}$ Frobenius angles: $\pm0.0154486843224$, $\pm0.247870790543$ Angle rank: $2$ (numerical) Number field: 4.0.1806444.1 Galois group: $D_{4}$ Jacobians: 10

This isogeny class is simple and geometrically simple.

## Newton polygon

This isogeny class is ordinary.

 $p$-rank: $2$ Slopes: $[0, 0, 1, 1]$

## Point counts

This isogeny class contains the Jacobians of 10 curves, and hence is principally polarizable:

• $y^2=22x^6+170x^5+134x^4+97x^3+41x^2+127x+87$
• $y^2=100x^6+154x^5+5x^4+116x^3+126x^2+92x+131$
• $y^2=12x^6+44x^5+171x^4+43x^3+167x+151$
• $y^2=32x^6+92x^5+85x^4+38x^3+32x^2+109x+59$
• $y^2=135x^6+151x^5+42x^4+27x^3+87x^2+53x+74$
• $y^2=130x^6+70x^5+169x^4+131x^3+72x^2+123x+131$
• $y^2=42x^6+132x^5+126x^4+24x^3+83x^2+103x+109$
• $y^2=115x^6+5x^5+71x^4+56x^3+115x^2+66x+17$
• $y^2=78x^6+87x^5+115x^4+24x^3+62x^2+31x+2$
• $y^2=126x^6+62x^5+33x^4+155x^3+7x^2+159x+5$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 22938 885360924 26801759572056 802360164759947136 24013778529438943026018 718709000431308768206228736 21510248278619325568734388129362 643780248519041816573257694970637824 19267699135292190124750962834299702512344 576662967575038684592778661076500570190273724

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 129 29581 5176368 895746145 154963702869 26808743828122 4637914106334393 802359175029023329 138808137837378759264 24013807852316095005061

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{173}$
 The endomorphism algebra of this simple isogeny class is 4.0.1806444.1.
All geometric endomorphisms are defined over $\F_{173}$.

## 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.bt_bgg $2$ (not in LMFDB)