# Properties

 Label 2.199.aby_bmz Base Field $\F_{199}$ Dimension $2$ Ordinary Yes $p$-rank $2$ Principally polarizable Yes Contains a Jacobian Yes

## Invariants

 Base field: $\F_{199}$ Dimension: $2$ L-polynomial: $1 - 50 x + 1013 x^{2} - 9950 x^{3} + 39601 x^{4}$ Frobenius angles: $\pm0.0191784432260$, $\pm0.218244196354$ Angle rank: $2$ (numerical) Number field: 4.0.1473600.1 Galois group: $D_{4}$ Jacobians: 6

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 6 curves, and hence is principally polarizable:

• $y^2=68x^6+167x^5+196x^4+114x^3+184x^2+112x+27$
• $y^2=18x^6+37x^5+13x^4+53x^3+185x^2+93x+183$
• $y^2=123x^6+27x^5+187x^4+74x^3+169x^2+128x+143$
• $y^2=105x^6+138x^5+79x^4+182x^3+163x^2+24x+50$
• $y^2=74x^6+105x^5+78x^4+188x^3+20x^2+169x+192$
• $y^2=85x^6+61x^5+2x^4+107x^3+157x^2+9x+174$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 30615 1549578225 62080988133060 2459368019623545225 97393678841553823350375 3856886653100608621553763600 152736580019093463529344791624535 6048521401857953932435504380335721225 239527496478819655379817919394917857743940 9485528389543214778777981047643795186450140625

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 150 39128 7877700 1568235268 312079605750 62103834737318 12358664056975050 2459374186584537028 489415464037206395100 97393677358656372840248

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
 The endomorphism algebra of this simple isogeny class is 4.0.1473600.1.
All geometric endomorphisms are defined over $\F_{199}$.

## 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 2.199.by_bmz $2$ (not in LMFDB)