# Properties

 Label 2.163.abs_bez 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: $1 - 44 x + 805 x^{2} - 7172 x^{3} + 26569 x^{4}$ Frobenius angles: $\pm0.101937801983$, $\pm0.218245318378$ Angle rank: $2$ (numerical) Number field: 4.0.422225.1 Galois group: $D_{4}$ Jacobians: 30

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 30 curves, and hence is principally polarizable:

• $y^2=57x^6+106x^5+111x^4+137x^3+31x^2+152x+62$
• $y^2=119x^6+133x^5+16x^4+15x^3+4x^2+42x+106$
• $y^2=120x^6+143x^5+31x^4+71x^3+148x^2+133x+5$
• $y^2=122x^6+35x^5+47x^4+155x^3+7x^2+57x+12$
• $y^2=49x^6+113x^5+85x^4+72x^3+20x^2+80x+59$
• $y^2=110x^6+102x^5+148x^4+68x^3+49x^2+45x+74$
• $y^2=125x^6+30x^5+22x^4+108x^3+77x^2+88x+94$
• $y^2=94x^6+133x^5+52x^4+17x^3+104x^2+34x+78$
• $y^2=11x^6+161x^5+135x^4+54x^3+156x^2+39x+149$
• $y^2=142x^6+55x^5+16x^4+138x^3+129x^2+140x+89$
• $y^2=87x^6+87x^5+140x^4+29x^3+118x^2+6x+147$
• $y^2=98x^6+99x^5+137x^4+61x^3+130x^2+48x+73$
• $y^2=88x^6+132x^5+65x^4+48x^3+43x^2+78x+147$
• $y^2=137x^6+50x^5+161x^4+133x^3+11x^2+148x+147$
• $y^2=149x^6+28x^5+78x^4+78x^3+3x^2+119x+156$
• $y^2=99x^6+59x^5+65x^4+57x^3+23x^2+101x+160$
• $y^2=116x^6+2x^5+33x^4+66x^3+105x^2+69x+105$
• $y^2=157x^6+70x^5+120x^4+70x^3+24x+120$
• $y^2=159x^6+129x^5+32x^4+100x^3+32x^2+88x+120$
• $y^2=33x^6+103x^5+20x^4+118x^3+70x^2+96x+67$
• $y^2=143x^6+25x^5+145x^4+132x^3+57x^2+104x+40$
• $y^2=38x^6+45x^5+49x^4+25x^3+135x^2+3x+119$
• $y^2=116x^6+123x^5+67x^4+123x^3+157x^2+128x+137$
• $y^2=118x^6+67x^5+39x^4+47x^3+99x^2+105x+139$
• $y^2=70x^6+96x^5+153x^4+34x^3+143x^2+27x+53$
• $y^2=27x^6+107x^5+58x^4+103x^3+103x^2+22x+161$
• $y^2=115x^6+104x^5+60x^4+159x^3+3x^2+68x+62$
• $y^2=11x^6+67x^5+30x^4+94x^3+41x^2+41x+129$
• $y^2=26x^6+53x^5+76x^4+47x^3+127x^2+37x+136$
• $y^2=34x^6+33x^5+36x^4+14x^3+95x^2+29x+20$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 20159 697319969 18753468073664 498335259273928121 13239713220513952313439 351764035348965653044576256 9346014921257106926288106394679 248314265737229287625651061316740329 6597461723742160079979377897507574286016 175287960539642206248789684544930545089716449

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 120 26244 4330308 705945540 115064288440 18755377433958 3057125300349576 498311414513788164 81224760533364674364 13239635967054514227364

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{163}$
 The endomorphism algebra of this simple isogeny class is 4.0.422225.1.
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_bez $2$ (not in LMFDB)