Invariants
| Base field: | $\F_{47}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 4 x + 47 x^{2} )^{2}$ |
| $1 - 8 x + 110 x^{2} - 376 x^{3} + 2209 x^{4}$ | |
| Frobenius angles: | $\pm0.405769133324$, $\pm0.405769133324$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $30$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1936$ | $5234944$ | $10883496976$ | $23795040096256$ | $52585297678744336$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $40$ | $2366$ | $104824$ | $4876350$ | $229284680$ | $10779130622$ | $506625617048$ | $23811300629374$ | $1119130411633768$ | $52599131333476286$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):
- $y^2=15 x^5+28 x^4+15 x^3+34 x^2+15 x+39$
- $y^2=25 x^6+43 x^5+6 x^4+9 x^3+37 x^2+29 x+1$
- $y^2=25 x^6+34 x^5+27 x^4+34 x^3+46 x^2+26 x+22$
- $y^2=23 x^6+17 x^5+45 x^4+x^3+35 x^2+8 x+32$
- $y^2=36 x^6+6 x^5+17 x^4+20 x^3+21 x^2+23 x+46$
- $y^2=15 x^6+13 x^5+14 x^4+17 x^3+14 x^2+13 x+15$
- $y^2=35 x^6+12 x^5+35 x^4+35 x^3+10 x^2+40 x+7$
- $y^2=25 x^6+9 x^4+9 x^2+25$
- $y^2=34 x^6+x^5+11 x^4+40 x^2+14 x+9$
- $y^2=46 x^6+3 x^5+3 x^4+12 x^3+45 x^2+39 x+38$
- $y^2=26 x^6+19 x^4+19 x^2+26$
- $y^2=39 x^6+5 x^5+20 x^4+4 x^3+31 x^2+6 x+20$
- $y^2=25 x^6+12 x^5+17 x^4+33 x^3+8 x^2+4 x+18$
- $y^2=40 x^6+13 x^5+2 x^4+24 x^3+12 x^2+5 x+13$
- $y^2=13 x^6+12 x^5+25 x^4+40 x^3+25 x^2+12 x+13$
- $y^2=16 x^6+7 x^5+38 x^4+27 x^3+13 x^2+31 x$
- $y^2=4 x^6+x^5+41 x^3+28 x^2+15 x+25$
- $y^2=24 x^6+16 x^5+32 x^4+43 x^3+6 x^2+3 x+23$
- $y^2=37 x^6+21 x^5+30 x^4+23 x^3+8 x^2+9 x+3$
- $y^2=10 x^6+5 x^4+5 x^2+10$
- $y^2=44 x^6+9 x^5+42 x^4+25 x^3+33 x^2+11 x+20$
- $y^2=35 x^6+44 x^5+25 x^4+34 x^3+25 x^2+44 x+35$
- $y^2=27 x^6+18 x^5+12 x^4+x^3+19 x^2+31 x+1$
- $y^2=35 x^6+17 x^5+6 x^4+43 x^3+6 x^2+17 x+35$
- $y^2=21 x^6+38 x^5+39 x^4+21 x^3+44 x^2+45 x+37$
- $y^2=44 x^6+16 x^5+7 x^4+15 x^3+32 x^2+25 x+11$
- $y^2=5 x^6+2 x^5+25 x^4+35 x^3+39 x^2+34 x+32$
- $y^2=35 x^6+34 x^5+8 x^3+30 x^2+15 x+15$
- $y^2=35 x^6+31 x^5+2 x^4+12 x^3+32 x^2+40 x+10$
- $y^2=28 x^6+37 x^5+6 x^4+13 x^3+7 x^2+26 x+35$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{47}$.
Endomorphism algebra over $\F_{47}$| The isogeny class factors as 1.47.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$ |
Base change
This is a primitive isogeny class.