Invariants
| Base field: | $\F_{47}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 20 x + 189 x^{2} - 940 x^{3} + 2209 x^{4}$ |
| Frobenius angles: | $\pm0.149015098474$, $\pm0.308396401021$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-298 -46 \sqrt{5}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $10$ |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1439$ | $4833601$ | $10833344576$ | $23833717988441$ | $52603054410179239$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $28$ | $2188$ | $104344$ | $4884276$ | $229362108$ | $10779226462$ | $506623280884$ | $23811293692068$ | $1119130556079688$ | $52599132692844828$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=22 x^6+40 x^5+16 x^4+13 x^3+6 x^2+19 x+38$
- $y^2=5 x^6+15 x^5+23 x^4+37 x^3+6 x^2+16 x+43$
- $y^2=40 x^6+36 x^5+x^4+8 x^3+45 x^2+21 x+5$
- $y^2=43 x^6+17 x^5+32 x^4+15 x^3+36 x^2+15 x+41$
- $y^2=22 x^6+41 x^5+23 x^4+20 x^3+46 x^2+41 x+16$
- $y^2=43 x^6+8 x^5+24 x^4+25 x^3+42 x^2+29 x+41$
- $y^2=10 x^6+28 x^5+17 x^4+25 x^3+28 x^2+10 x+11$
- $y^2=38 x^6+28 x^5+3 x^4+44 x^3+15 x^2+45 x+9$
- $y^2=35 x^6+46 x^5+30 x^4+33 x^3+14 x^2+7 x+38$
- $y^2=30 x^6+40 x^5+24 x^4+7 x^3+44 x^2+8 x+45$
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 \(\Q(\sqrt{-298 -46 \sqrt{5}})\). |
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_hh | $2$ | (not in LMFDB) |