# Properties

 Label 2.163.abu_bgr 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: $1 - 46 x + 849 x^{2} - 7498 x^{3} + 26569 x^{4}$ Frobenius angles: $\pm0.0259491891705$, $\pm0.202261752526$ Angle rank: $2$ (numerical) Number field: 4.0.571968.5 Galois group: $D_{4}$ Jacobians: 4

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 4 curves, and hence is principally polarizable:

• $y^2=8x^6+60x^5+38x^4+7x^3+81x^2+108x+29$
• $y^2=59x^6+124x^5+52x^4+96x^3+17x^2+58x+32$
• $y^2=11x^6+125x^5+136x^4+118x^3+43x^2+119x+109$
• $y^2=87x^6+142x^5+68x^4+58x^3+96x^2+50x+109$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 19875 694889625 18743818270500 498306840625745625 13239642313044741961875 351763871842138011168474000 9346014544332082720784570524875 248314264825521691677280610111075625 6597461721457477517793056403453748426500 175287960534032556327597431329248589783685625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 118 26152 4328080 705905284 115063672198 18755368716094 3057125177055634 498311412684194116 81224760505236766720 13239635966630813006632

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{163}$
 The endomorphism algebra of this simple isogeny class is 4.0.571968.5.
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_bgr $2$ (not in LMFDB)