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, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $30615$ | $1549578225$ | $62080988133060$ | $2459368019623545225$ | $97393678841553823350375$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $150$ | $39128$ | $7877700$ | $1568235268$ | $312079605750$ | $62103834737318$ | $12358664056975050$ | $2459374186584537028$ | $489415464037206395100$ | $97393677358656372840248$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=68x^6+167x^5+196x^4+114x^3+184x^2+112x+27$
- $y^2=18x^6+37x^5+13x^4+53x^3+185x^2+93x+183$
- $y^2=123x^6+27x^5+187x^4+74x^3+169x^2+128x+143$
- $y^2=105x^6+138x^5+79x^4+182x^3+163x^2+24x+50$
- $y^2=74x^6+105x^5+78x^4+188x^3+20x^2+169x+192$
- $y^2=85x^6+61x^5+2x^4+107x^3+157x^2+9x+174$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{199}$.
Endomorphism algebra over $\F_{199}$| The endomorphism algebra of this simple isogeny class is 4.0.1473600.1. |
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) |