# Properties

 Label 2.163.abu_bgs 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: $1 - 46 x + 850 x^{2} - 7498 x^{3} + 26569 x^{4}$ Frobenius angles: $\pm0.0486964346780$, $\pm0.197735354640$ Angle rank: $2$ (numerical) Number field: 4.0.83600.1 Galois group: $D_{4}$ Jacobians: 12

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 12 curves, and hence is principally polarizable:

• $y^2=81x^6+57x^5+65x^4+123x^3+130x^2+47x+73$
• $y^2=121x^6+15x^5+159x^3+74x^2+63x+137$
• $y^2=45x^6+30x^5+88x^4+48x^3+113x^2+25x+126$
• $y^2=28x^6+51x^5+139x^4+123x^3+161x^2+63x+116$
• $y^2=55x^6+26x^5+154x^4+103x^3+119x^2+24x+94$
• $y^2=92x^6+28x^5+41x^4+132x^3+139x^2+77x+59$
• $y^2=55x^6+72x^5+87x^4+134x^3+40x^2+74x+46$
• $y^2=153x^6+91x^5+117x^4+54x^3+145x^2+87x+160$
• $y^2=139x^6+110x^5+77x^4+92x^3+74x^2+14x+103$
• $y^2=71x^6+42x^5+136x^4+29x^3+30x^2+130x+138$
• $y^2=144x^6+84x^5+125x^4+95x^3+42x^2+162x+8$
• $y^2=11x^6+155x^5+75x^4+45x^3+82x^2+58x+29$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 19876 694944464 18744416964244 498310417152102656 13239657662621701104916 351763924701610325593522256 9346014698771740603393140190756 248314265221535860602831836758052864 6597461722372755883485648672745615183876 175287960535997057480030111929678805751622224

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 118 26154 4328218 705910350 115063805598 18755371534458 3057125227573570 498311413478906334 81224760516505231654 13239635966779193306314

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{163}$
 The endomorphism algebra of this simple isogeny class is 4.0.83600.1.
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.bu_bgs $2$ (not in LMFDB)