Invariants
| Base field: | $\F_{47}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 12 x + 47 x^{2} )^{2}$ |
| $1 - 24 x + 238 x^{2} - 1128 x^{3} + 2209 x^{4}$ | |
| Frobenius angles: | $\pm0.160736311100$, $\pm0.160736311100$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $10$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1296$ | $4665600$ | $10771948944$ | $23830018560000$ | $52610466617920656$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $24$ | $2110$ | $103752$ | $4883518$ | $229394424$ | $10779628030$ | $506625750312$ | $23811298823038$ | $1119130495435224$ | $52599131932239550$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=31 x^6+26 x^4+26 x^2+31$
- $y^2=35 x^6+6 x^4+6 x^2+35$
- $y^2=44 x^6+12 x^5+8 x^4+18 x^3+12 x^2+27 x+31$
- $y^2=16 x^6+22 x^5+24 x^4+15 x^3+24 x^2+22 x+16$
- $y^2=5 x^6+20 x^5+22 x^4+42 x^3+46 x^2+35 x+45$
- $y^2=44 x^6+32 x^5+37 x^4+13 x^3+37 x^2+32 x+44$
- $y^2=40 x^6+20 x^5+15 x^4+16 x^3+40 x^2+43 x+10$
- $y^2=38 x^6+26 x^5+28 x^4+10 x^3+28 x^2+26 x+38$
- $y^2=4 x^6+20 x^4+20 x^2+4$
- $y^2=22 x^6+19 x^5+24 x^4+4 x^3+x^2+29 x+35$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{47}$.
Endomorphism algebra over $\F_{47}$| The isogeny class factors as 1.47.am 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$ |
Base change
This is a primitive isogeny class.