# Properties

 Label 2.163.abu_bgu 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: $1 - 46 x + 852 x^{2} - 7498 x^{3} + 26569 x^{4}$ Frobenius angles: $\pm0.0800016625006$, $\pm0.186669006544$ Angle rank: $2$ (numerical) Number field: 4.0.1159488.4 Galois group: $D_{4}$ Jacobians: 16

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 16 curves, and hence is principally polarizable:

• $y^2=141x^6+87x^5+38x^4+73x^3+38x^2+131x+65$
• $y^2=140x^6+56x^5+92x^4+94x^3+154x^2+118x+153$
• $y^2=87x^6+59x^5+44x^4+112x^3+144x^2+98x+114$
• $y^2=94x^6+6x^5+130x^4+75x^3+74x^2+14x+2$
• $y^2=75x^6+138x^5+19x^4+29x^3+56x^2+37x+41$
• $y^2=56x^6+52x^5+112x^4+101x^3+149x^2+72x+130$
• $y^2=56x^6+62x^5+77x^4+32x^3+48x^2+124x+67$
• $y^2=56x^6+46x^5+134x^4+141x^3+107x^2+159x+91$
• $y^2=58x^6+128x^5+87x^4+162x^3+117x^2+89x+120$
• $y^2=148x^6+127x^5+109x^4+66x^3+29x^2+71x+11$
• $y^2=99x^6+67x^5+107x^4+158x^3+33x^2+30x+73$
• $y^2=86x^6+122x^5+42x^4+156x^3+78x^2+117x+58$
• $y^2=31x^6+8x^5+149x^4+105x^3+91x^2+67x+120$
• $y^2=108x^6+106x^5+53x^4+90x^3+23x^2+121x+45$
• $y^2=48x^6+95x^5+139x^4+105x^3+57x^2+124x+92$
• $y^2=89x^6+91x^5+136x^4+58x^3+126x^2+72x+45$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 19878 695054148 18745614367038 498317561750990544 13239688202999662862238 351764028851636602829476836 9346014996759719715956816285574 248314265953992534556905732024972288 6597461723930601469548592797805709926422 175287960538842415394931453502414310626585188

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 118 26158 4328494 705920470 115064071018 18755377087534 3057125325046834 498311414948783710 81224760535684673686 13239635966994105402238

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{163}$
 The endomorphism algebra of this simple isogeny class is 4.0.1159488.4.
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_bgu $2$ (not in LMFDB)