Invariants
| Base field: | $\F_{47}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 8 x + 104 x^{2} - 376 x^{3} + 2209 x^{4}$ |
| Frobenius angles: | $\pm0.344117799529$, $\pm0.463927535095$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-166 +8 \sqrt{6}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $28$ |
| Isomorphism classes: | 28 |
| Cyclic group of points: | yes |
This isogeny class is simple and 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1930$ | $5207140$ | $10868434090$ | $23800066512400$ | $52590966492068650$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $40$ | $2354$ | $104680$ | $4877382$ | $229309400$ | $10779173426$ | $506623732760$ | $23811287610238$ | $1119130479913960$ | $52599132514467314$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):
- $y^2=38 x^6+8 x^4+21 x^3+27 x^2+17 x+2$
- $y^2=23 x^6+30 x^5+35 x^4+12 x^3+2 x^2+19 x+11$
- $y^2=27 x^6+17 x^5+24 x^4+20 x^3+25 x^2+22 x+10$
- $y^2=3 x^6+16 x^5+42 x^4+15 x^3+26 x^2+9 x+3$
- $y^2=31 x^6+15 x^5+26 x^4+32 x^3+36 x^2+36 x+1$
- $y^2=26 x^6+30 x^5+45 x^4+26 x^3+14 x^2+40 x+28$
- $y^2=27 x^6+15 x^5+39 x^4+25 x^3+4 x^2+25 x+35$
- $y^2=13 x^6+11 x^5+18 x^4+30 x^3+44 x^2+5 x+18$
- $y^2=23 x^5+41 x^4+30 x^3+45 x^2+9 x+39$
- $y^2=2 x^6+18 x^5+3 x^4+8 x^3+29 x^2+5 x+17$
- $y^2=22 x^6+11 x^5+43 x^4+6 x^3+x^2+34 x+39$
- $y^2=2 x^6+32 x^5+9 x^4+31 x^3+19 x^2+28 x+45$
- $y^2=23 x^6+38 x^5+44 x^4+24 x^3+23 x^2+15 x+27$
- $y^2=x^6+38 x^5+42 x^4+18 x^3+31 x^2+28 x+4$
- $y^2=33 x^6+21 x^5+9 x^4+8 x^3+20 x^2+37 x+31$
- $y^2=11 x^6+5 x^5+9 x^4+9 x^3+3 x^2+7 x+2$
- $y^2=27 x^6+9 x^5+15 x^2+45 x+8$
- $y^2=41 x^6+35 x^5+43 x^4+30 x^3+22 x^2+28 x+34$
- $y^2=28 x^6+12 x^5+34 x^4+18 x^3+x^2+41 x+7$
- $y^2=44 x^6+46 x^5+4 x^4+45 x^3+23 x^2+41 x+44$
- $y^2=12 x^6+46 x^5+30 x^4+13 x^3+20 x^2+21 x$
- $y^2=36 x^6+4 x^5+10 x^4+32 x^3+37 x^2+8 x+18$
- $y^2=x^6+7 x^5+18 x^4+40 x^3+17 x^2+19 x+9$
- $y^2=31 x^6+29 x^5+8 x^4+30 x^3+28 x^2+36 x+19$
- $y^2=24 x^6+23 x^5+27 x^4+30 x^3+33 x^2+17 x+6$
- $y^2=33 x^6+7 x^5+2 x^4+28 x^3+44 x^2+22 x+2$
- $y^2=41 x^6+29 x^5+6 x^4+46 x^3+28 x^2+40 x+43$
- $y^2=31 x^6+40 x^5+20 x^4+37 x^3+24 x^2+6 x+31$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{47}$.
Endomorphism algebra over $\F_{47}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-166 +8 \sqrt{6}})\). |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.47.i_ea | $2$ | (not in LMFDB) |