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: | $\pm0.239834262915$, $\pm0.239834262915$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $8$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 19$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1444$ | $4857616$ | $10864726756$ | $23854081156096$ | $52610394137115364$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $28$ | $2198$ | $104644$ | $4888446$ | $229394108$ | $10779294422$ | $506621603684$ | $23811267776638$ | $1119130355540188$ | $52599131947805078$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=23 x^6+40 x^5+46 x^4+37 x^3+14 x^2+46 x+13$
- $y^2=18 x^6+44 x^4+44 x^2+18$
- $y^2=43 x^6+7 x^5+33 x^4+19 x^3+33 x^2+7 x+43$
- $y^2=10 x^6+10 x^5+3 x^4+32 x^3+19 x^2+7 x+46$
- $y^2=22 x^6+7 x^4+42 x^3+44 x^2+5 x+31$
- $y^2=15 x^6+8 x^5+5 x^4+28 x^3+26 x^2+2 x+43$
- $y^2=4 x^6+42 x^5+46 x^4+29 x^3+11 x^2+23 x+19$
- $y^2=10 x^6+3 x^5+5 x^4+46 x^3+2 x^2+13 x+28$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{47}$.
Endomorphism algebra over $\F_{47}$| The isogeny class factors as 1.47.ak 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-22}) \)$)$ |
Base change
This is a primitive isogeny class.