Invariants
| Base field: | $\F_{47}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 6 x - 11 x^{2} - 282 x^{3} + 2209 x^{4}$ |
| Frobenius angles: | $\pm0.0224970475152$, $\pm0.689163714182$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-38})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 30 |
| Cyclic group of points: | yes |
This isogeny class is simple but not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1911$ | $4752657$ | $10649001636$ | $23806139708169$ | $52593792020866551$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $42$ | $2152$ | $102564$ | $4878628$ | $229321722$ | $10778836822$ | $506623079382$ | $23811278013316$ | $1119130365459708$ | $52599132319377832$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=20 x^6+46 x^5+13 x^4+43 x^3+36 x^2+27 x+20$
- $y^2=31 x^6+14 x^5+3 x^4+11 x^3+22 x^2+31 x+31$
- $y^2=17 x^6+28 x^5+6 x^4+42 x^3+27 x^2+27 x+17$
- $y^2=46 x^6+23 x^5+5 x^4+26 x^3+16 x^2+18 x+46$
- $y^2=38 x^6+x^5+30 x^4+12 x^3+31 x^2+39 x+38$
- $y^2=20 x^6+22 x^5+43 x^3+2 x^2+4 x+20$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{47^{3}}$.
Endomorphism algebra over $\F_{47}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-38})\). |
| The base change of $A$ to $\F_{47^{3}}$ is 1.103823.ayg 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-38}) \)$)$ |
Base change
This is a primitive isogeny class.