# Properties

 Label 2.163.abs_bew 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: $1 - 44 x + 802 x^{2} - 7172 x^{3} + 26569 x^{4}$ Frobenius angles: $\pm0.0750186375807$, $\pm0.229660133188$ Angle rank: $2$ (numerical) Number field: 4.0.161792.1 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:

• $y^2=33x^6+2x^5+47x^4+143x^3+144x^2+13x+61$
• $y^2=62x^6+75x^5+28x^4+77x^3+21x^2+130x+5$
• $y^2=124x^6+143x^5+102x^4+41x^3+20x^2+142x+40$
• $y^2=127x^6+68x^5+30x^4+91x^3+86x^2+116x+51$
• $y^2=3x^6+96x^5+159x^4+122x^3+110x^2+128x+104$
• $y^2=19x^6+116x^5+152x^4+75x^3+75x^2+121x+131$
• $y^2=45x^6+80x^5+6x^4+88x^2+44x+2$
• $y^2=71x^6+107x^5+149x^4+104x^3+52x^2+54x+92$
• $y^2=98x^6+88x^5+157x^4+22x^3+162x^2+18x+70$
• $y^2=117x^6+82x^5+114x^4+162x^3+105x^2+66x+47$
• $y^2=141x^6+133x^5+39x^4+106x^3+162x^2+156x+44$
• $y^2=151x^6+15x^5+52x^4+67x^3+113x^2+131x+53$
• $y^2=30x^6+75x^5+158x^4+143x^3+156x^2+95x+75$
• $y^2=156x^6+155x^5+80x^4+57x^3+78x^2+119x+88$
• $y^2=120x^6+60x^5+37x^4+139x^3+95x^2+82x+110$
• $y^2=75x^6+106x^5+122x^4+86x^3+20x^2+155x+53$
• $y^2=98x^6+94x^5+40x^4+137x^3+141x^2+155x+158$
• $y^2=11x^6+48x^5+162x^4+50x^3+74x^2+81x+110$
• $y^2=7x^6+55x^5+33x^4+150x^3+47x^2+91x+81$
• $y^2=77x^6+25x^5+94x^4+19x^3+16x^2+4x+11$
• $y^2=129x^6+149x^5+113x^4+106x^3+21x^2+12x+15$
• $y^2=38x^6+98x^5+150x^4+140x^3+89x^2+134x+116$
• $y^2=56x^6+82x^5+81x^4+72x^3+109x^2+123x+114$
• $y^2=158x^6+41x^5+34x^4+129x^3+70x^2+71x+24$
• $y^2=28x^6+108x^5+132x^3+40x^2+58x+23$
• $y^2=141x^6+55x^5+32x^4+21x^3+66x^2+6x+118$
• $y^2=69x^6+17x^5+55x^4+41x^3+45x^2+160x+22$
• $y^2=39x^6+6x^5+46x^4+20x^3+22x^2+84x+92$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 20156 697155728 18751750366172 498325665071335424 13239675856885541673276 351763923419451839435862416 9346014655004292683495701109468 248314265243853003677588645263474688 6597461723115701991549958461823964276156 175287960539499484493977236503820760258297488

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 120 26238 4329912 705931950 115063963720 18755371466094 3057125213257032 498311413523691870 81224760525652025112 13239635967043734342238

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{163}$
 The endomorphism algebra of this simple isogeny class is 4.0.161792.1.
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.bs_bew $2$ (not in LMFDB)