# Properties

 Label 2.173.abt_bgk 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: $1 - 45 x + 842 x^{2} - 7785 x^{3} + 29929 x^{4}$ Frobenius angles: $\pm0.0683613928513$, $\pm0.237830868262$ Angle rank: $2$ (numerical) Number field: 4.0.208444.1 Galois group: $D_{4}$ Jacobians: 18

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 18 curves, and hence is principally polarizable:

• $y^2=3x^6+24x^5+138x^4+20x^3+75x^2+131x+34$
• $y^2=95x^6+79x^5+80x^4+103x^3+165x^2+72x+2$
• $y^2=91x^6+33x^5+64x^4+132x^3+3x^2+45x+45$
• $y^2=95x^6+85x^5+109x^4+14x^3+129x^2+93x+104$
• $y^2=114x^6+144x^5+143x^4+67x^3+58x^2+149x+59$
• $y^2=166x^6+40x^5+132x^4+47x^3+123x^2+147x+98$
• $y^2=127x^6+59x^5+101x^4+77x^3+134x^2+12x+45$
• $y^2=91x^6+148x^5+166x^4+153x^3+109x^2+36x+65$
• $y^2=101x^6+21x^5+21x^4+3x^3+107x^2+58x+88$
• $y^2=106x^6+92x^5+120x^4+135x^3+28x^2+55x+64$
• $y^2=144x^6+65x^5+51x^4+58x^3+6x^2+89x+154$
• $y^2=3x^6+149x^5+15x^4+37x^3+84x^2+10x+145$
• $y^2=149x^6+5x^5+142x^4+109x^3+8x^2+52x+101$
• $y^2=158x^6+43x^5+164x^4+17x^3+94x^2+156x+64$
• $y^2=129x^6+122x^5+130x^4+7x^3+122x^2+160x+166$
• $y^2=170x^6+137x^5+44x^4+100x^3+130x^2+4x+49$
• $y^2=40x^6+5x^5+112x^4+14x^3+141x^2+104x+101$
• $y^2=26x^6+172x^5+54x^4+29x^3+108x^2+11x+51$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 22942 885607084 26804559443944 802377149287432896 24013850773800021375382 718709241032789744198738176 21510248944260013690520681894998 643780250125101329523931652675261184 19267699138904804177151254551529223616936 576662967583401714179572636406416912188109324

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 129 29589 5176908 895765105 154964169069 26808752802858 4637914249855953 802359177030694849 138808137863404712124 24013807852664354208189

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{173}$
 The endomorphism algebra of this simple isogeny class is 4.0.208444.1.
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_bgk $2$ (not in LMFDB)