# Properties

 Label 2.173.abt_bgm 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: $1 - 45 x + 844 x^{2} - 7785 x^{3} + 29929 x^{4}$ Frobenius angles: $\pm0.0850578884700$, $\pm0.231907856828$ Angle rank: $2$ (numerical) Number field: 4.0.1007325.1 Galois group: $D_{4}$ Jacobians: 80

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 80 curves, and hence is principally polarizable:

• $y^2=74x^6+69x^5+2x^4+119x^3+147x^2+163x+74$
• $y^2=60x^6+18x^5+116x^4+84x^3+120x^2+11x+80$
• $y^2=77x^6+147x^5+149x^4+132x^3+73x^2+7x+10$
• $y^2=19x^6+24x^5+160x^4+41x^3+84x^2+99x+81$
• $y^2=110x^6+11x^5+62x^4+15x^3+74x^2+39x+43$
• $y^2=21x^6+81x^5+133x^4+x^3+62x^2+100x+101$
• $y^2=155x^6+169x^5+113x^4+78x^3+140x^2+159x+66$
• $y^2=70x^6+37x^5+149x^4+3x^3+169x^2+75x+74$
• $y^2=28x^6+152x^5+78x^4+143x^3+165x^2+157x+15$
• $y^2=27x^6+128x^5+127x^4+86x^3+120x^2+163x+66$
• $y^2=97x^6+163x^5+154x^4+102x^3+77x^2+169x+50$
• $y^2=135x^6+49x^5+143x^4+56x^3+88x^2+59x+77$
• $y^2=32x^6+16x^5+96x^4+22x^3+130x^2+159x+85$
• $y^2=12x^6+92x^5+140x^4+26x^3+17x^2+157x+41$
• $y^2=127x^6+106x^5+18x^4+47x^3+105x^2+165x+74$
• $y^2=123x^6+33x^5+15x^4+85x^3+108x^2+39x+34$
• $y^2=28x^6+71x^5+44x^4+8x^3+51x^2+91x+166$
• $y^2=163x^6+127x^5+88x^4+132x^3+128x^2+168x+158$
• $y^2=147x^6+49x^5+164x^4+60x^3+2x^2+81x+121$
• $y^2=17x^6+152x^5+170x^4+21x^3+142x^2+35x+31$
• and 60 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 22944 885730176 26805959410176 802385620086007296 24013886477599203154464 718709357094432210483388416 21510249247294654495373909580576 643780250765437794358882532428339200 19267699139977511921507096561075469832704 576662967584750627470942998564295472114791296

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 129 29593 5177178 895774561 154964399469 26808757132102 4637914315194513 802359177828761953 138808137871132700994 24013807852720526611153

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{173}$
 The endomorphism algebra of this simple isogeny class is 4.0.1007325.1.
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_bgm $2$ (not in LMFDB)