# Properties

 Label 2.199.aca_bpg 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: $1 - 52 x + 1072 x^{2} - 10348 x^{3} + 39601 x^{4}$ Frobenius angles: $\pm0.0759470812814$, $\pm0.163199525114$ Angle rank: $2$ (numerical) Number field: 4.0.545024.2 Galois group: $D_{4}$ Jacobians: 8

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 8 curves, and hence is principally polarizable:

• $y^2=7x^6+157x^5+163x^4+157x^3+101x^2+13x+98$
• $y^2=160x^6+180x^5+105x^4+60x^3+30x^2+43x+147$
• $y^2=135x^6+138x^5+83x^4+18x^3+57x^2+52x+187$
• $y^2=55x^6+141x^5+12x^4+188x^3+186x^2+48x+50$
• $y^2=120x^6+33x^5+129x^4+93x^3+39x^2+146x+9$
• $y^2=90x^6+106x^5+28x^4+135x^3+105x^2+18x+6$
• $y^2=175x^6+151x^5+170x^4+122x^3+186x^2+51x+144$
• $y^2=73x^6+44x^5+113x^4+31x^3+147x^2+81x+196$

 $r$ 1 2 3 4 5 6 7 8 9 10 $A(\F_{q^r})$ 30274 1546214276 62069024885122 2459359746395334416 97393840761067648525474 3856887858022279786406442884 152736585515030250461136549302434 6048521421058804970950352584136474624 239527496532809355490184618063565954131842 9485528389664127202401972176748933997315100676

 $r$ 1 2 3 4 5 6 7 8 9 10 $C(\F_{q^r})$ 148 39042 7876180 1568229990 312080124588 62103854139042 12358664501678188 2459374194391747134 489415464147521054260 97393677359897854083202

## Decomposition and endomorphism algebra

Endomorphism algebra over $\F_{199}$
 The endomorphism algebra of this simple isogeny class is 4.0.545024.2.
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.ca_bpg $2$ (not in LMFDB)