Invariants
| Base field: | $\F_{47}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 10 x + 47 x^{2} )^{2}$ |
| $1 + 20 x + 194 x^{2} + 940 x^{3} + 2209 x^{4}$ | |
| Frobenius angles: | $\pm0.760165737085$, $\pm0.760165737085$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $8$ |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3364$ | $4857616$ | $10694455396$ | $23854081156096$ | $52587872457329764$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $68$ | $2198$ | $103004$ | $4888446$ | $229295908$ | $10779294422$ | $506624637244$ | $23811267776638$ | $1119130590665348$ | $52599131947805078$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=21 x^6+12 x^5+42 x^4+44 x^3+23 x^2+42 x+18$
- $y^2=13 x^6+37 x^4+37 x^2+13$
- $y^2=18 x^6+39 x^5+16 x^4+32 x^3+16 x^2+39 x+18$
- $y^2=2 x^6+2 x^5+10 x^4+44 x^3+32 x^2+39 x+28$
- $y^2=42 x^6+39 x^4+46 x^3+37 x^2+x+25$
- $y^2=28 x^6+40 x^5+25 x^4+46 x^3+36 x^2+10 x+27$
- $y^2=20 x^6+22 x^5+42 x^4+4 x^3+8 x^2+21 x+1$
- $y^2=2 x^6+10 x^5+x^4+28 x^3+38 x^2+12 x+15$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{47}$.
Endomorphism algebra over $\F_{47}$| The isogeny class factors as 1.47.k 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-22}) \)$)$ |
Base change
This is a primitive isogeny class.