# Properties

 Label 2.173.abt_bgp 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: $1 - 45 x + 847 x^{2} - 7785 x^{3} + 29929 x^{4}$ Frobenius angles: $\pm0.108541149926$, $\pm0.221142714353$ Angle rank: $2$ (numerical) Number field: 4.0.9679509.2 Galois group: $D_{4}$ Jacobians: 28

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 28 curves, and hence is principally polarizable:

• $y^2=20x^6+153x^5+72x^4+83x^3+167x^2+65x+134$
• $y^2=164x^6+88x^5+32x^4+13x^3+115x^2+115x+58$
• $y^2=148x^6+132x^5+75x^4+70x^3+19x^2+25x+164$
• $y^2=125x^6+61x^5+131x^4+77x^3+53x^2+117x+145$
• $y^2=63x^6+165x^5+136x^4+19x^3+96x^2+50x+145$
• $y^2=67x^6+5x^5+53x^4+133x^2+64x+57$
• $y^2=132x^6+67x^5+80x^4+35x^3+138x^2+4x+20$
• $y^2=43x^6+105x^5+4x^4+7x^3+45x^2+155x+147$
• $y^2=157x^6+155x^5+103x^4+106x^3+53x^2+7x+70$
• $y^2=125x^6+71x^5+124x^4+159x^3+99x^2+18x+162$
• $y^2=81x^6+66x^5+40x^4+19x^3+134x^2+112x+75$
• $y^2=135x^6+4x^5+93x^4+145x^3+163x^2+121x+50$
• $y^2=97x^6+154x^5+25x^4+25x^3+51x^2+8x+25$
• $y^2=39x^6+51x^5+87x^4+13x^3+80x^2+77x+104$
• $y^2=90x^6+58x^5+119x^4+55x^3+20x^2+112x+75$
• $y^2=18x^6+25x^5+104x^4+52x^3+148x^2+127x+143$
• $y^2=92x^6+153x^5+112x^4+36x^3+46x^2+62x+167$
• $y^2=76x^6+28x^5+32x^4+68x^3+34x^2+124x+123$
• $y^2=111x^6+119x^5+138x^4+73x^3+138x^2+85x+69$
• $y^2=26x^6+38x^5+122x^4+12x^3+48x^2+56x+33$
• $y^2=144x^6+104x^5+138x^4+18x^3+104x^2+10x+113$
• $y^2=112x^6+122x^5+99x^4+127x^3+42x^2+145x+161$
• $y^2=167x^6+164x^5+108x^4+30x^3+2x^2+38x+82$
• $y^2=8x^6+5x^5+35x^4+24x^3+140x^2+63x+156$
• $y^2=133x^6+142x^5+121x^4+12x^3+135x^2+134x+168$
• $y^2=141x^6+51x^5+153x^4+138x^3+82x^2+63x+111$
• $y^2=170x^6+109x^5+27x^4+171x^3+126x^2+90x+29$
• $y^2=72x^6+94x^5+57x^4+37x^3+126x^2+25x+162$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 22947 885914829 26808059397519 802398299452863381 24013939510321981217232 718709525895262422149398701 21510249664811714400329303002863 643780251525355113091705275540554469 19267699140696531753532464600346948317411 576662967583435189927753340573121418991188224

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 129 29599 5177583 895788715 154964741694 26808763428583 4637914405217103 802359178775865619 138808137876312655449 24013807852665748229014

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{173}$
 The endomorphism algebra of this simple isogeny class is 4.0.9679509.2.
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_bgp $2$ (not in LMFDB)