Invariants
| Base field: | $\F_{47}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 6 x - 11 x^{2} + 282 x^{3} + 2209 x^{4}$ |
| Frobenius angles: | $\pm0.310836285818$, $\pm0.977502952485$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2487$ | $4752657$ | $10910638116$ | $23806139708169$ | $52604473076579127$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $54$ | $2152$ | $105084$ | $4878628$ | $229368294$ | $10778836822$ | $506623161546$ | $23811278013316$ | $1119130580745828$ | $52599132319377832$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=6 x^6+42 x^5+18 x^4+27 x^3+39 x^2+41 x+6$
- $y^2=14 x^6+23 x^5+15 x^4+8 x^3+16 x^2+14 x+14$
- $y^2=41 x^6+15 x^5+20 x^4+46 x^3+43 x^2+43 x+41$
- $y^2=28 x^6+14 x^5+x^4+24 x^3+22 x^2+13 x+28$
- $y^2=2 x^6+5 x^5+9 x^4+13 x^3+14 x^2+7 x+2$
- $y^2=6 x^6+16 x^5+27 x^3+10 x^2+20 x+6$
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.yg 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-38}) \)$)$ |
Base change
This is a primitive isogeny class.