Invariants
| Base field: | $\F_{47}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 20 x + 191 x^{2} - 940 x^{3} + 2209 x^{4}$ |
| Frobenius angles: | $\pm0.173159062413$, $\pm0.293971021494$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.867600.7 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1441$ | $4843201$ | $10845894004$ | $23841920997961$ | $52606127796416161$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $28$ | $2192$ | $104464$ | $4885956$ | $229375508$ | $10779268358$ | $506622839324$ | $23811285721348$ | $1119130490464048$ | $52599132381264272$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=13 x^6+38 x^5+29 x^4+40 x^3+18 x^2+19 x+10$
- $y^2=20 x^6+7 x^5+34 x^4+11 x^3+20 x^2+5 x+35$
- $y^2=34 x^6+8 x^5+45 x^4+19 x^3+37 x^2+31 x+8$
- $y^2=3 x^6+34 x^5+22 x^4+4 x^3+42 x^2+36 x+29$
- $y^2=19 x^6+22 x^5+3 x^4+44 x^3+25 x^2+42 x+26$
- $y^2=23 x^6+24 x^5+3 x^4+24 x^3+6 x^2+x+39$
- $y^2=13 x^6+31 x^5+22 x^4+45 x^3+27 x^2+4 x+40$
- $y^2=11 x^6+16 x^5+36 x^4+3 x^3+24 x^2+35$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{47}$.
Endomorphism algebra over $\F_{47}$| The endomorphism algebra of this simple isogeny class is 4.0.867600.7. |
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 |
|---|---|---|
| 2.47.u_hj | $2$ | (not in LMFDB) |