# Properties

 Label 2.163.abr_bdy Base Field $\F_{163}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{163}$ Dimension: $2$ L-polynomial: $1 - 43 x + 778 x^{2} - 7009 x^{3} + 26569 x^{4}$ Frobenius angles: $\pm0.0815159038334$, $\pm0.245687237918$ Angle rank: $2$ (numerical) Number field: 4.0.5575877.1 Galois group: $D_{4}$ Jacobians: 48

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 48 curves, and hence is principally polarizable:

• $y^2=90x^6+143x^5+147x^4+158x^3+30x^2+149x+82$
• $y^2=41x^6+39x^5+123x^4+27x^3+34x^2+13x+86$
• $y^2=23x^6+84x^5+124x^4+101x^3+81x^2+91x+74$
• $y^2=35x^6+41x^5+45x^4+93x^3+25x^2+61x+19$
• $y^2=78x^6+58x^5+63x^4+34x^3+105x^2+47x+20$
• $y^2=63x^6+139x^5+118x^4+150x^3+28x^2+46x+94$
• $y^2=80x^6+6x^5+78x^4+40x^3+24x^2+29x+31$
• $y^2=52x^6+33x^5+9x^4+53x^3+77x^2+5x+127$
• $y^2=58x^6+25x^5+35x^4+24x^3+124x^2+106x+54$
• $y^2=95x^6+78x^5+102x^4+37x^3+114x^2+92x+72$
• $y^2=39x^6+157x^5+17x^4+97x^3+34x^2+119x+27$
• $y^2=141x^6+95x^5+154x^4+70x^3+41x^2+10x+17$
• $y^2=18x^6+148x^5+3x^4+39x^3+9x^2+142x+104$
• $y^2=29x^6+39x^5+55x^4+121x^3+33x^2+73x+20$
• $y^2=158x^6+126x^5+110x^4+61x^3+53x^2+87x+123$
• $y^2=159x^6+61x^5+84x^4+116x^3+44x^2+8x+11$
• $y^2=143x^6+139x^4+160x^3+17x^2+123x+11$
• $y^2=156x^6+160x^5+138x^4+10x^3+21x^2+80x+51$
• $y^2=32x^6+115x^5+25x^4+95x^3+110x^2+156x+144$
• $y^2=56x^6+5x^5+144x^4+42x^3+100x^2+158x+8$
• and 28 more

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 20296 698182400 18754624907488 498329377601408000 13239672427012164614296 351763895635251518279782400 9346014599521176454508749826872 248314265263904043542485602622976000 6597461723609103493431153208241146815712 175287960541296716944071312186371191860192000

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 121 26277 4330576 705937209 115063933911 18755369984694 3057125195108245 498311413563929841 81224760531726545968 13239635967179480706757

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{163}$
 The endomorphism algebra of this simple isogeny class is 4.0.5575877.1.
All geometric endomorphisms are defined over $\F_{163}$.

## 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 2.163.br_bdy $2$ (not in LMFDB)