Invariants
| Base field: | $\F_{47}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 20 x + 188 x^{2} - 940 x^{3} + 2209 x^{4}$ |
| Frobenius angles: | $\pm0.137629815230$, $\pm0.314369396380$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.2490624.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Cyclic group of points: | yes |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $1$ |
| Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1438$ | $4828804$ | $10827071566$ | $23829587598736$ | $52601448964687918$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $28$ | $2186$ | $104284$ | $4883430$ | $229355108$ | $10779198122$ | $506623384484$ | $23811296391934$ | $1119130579584028$ | $52599132822374186$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=46 x^6+26 x^5+20 x^4+31 x^3+35 x^2+44 x+44$
- $y^2=31 x^6+19 x^5+20 x^4+15 x^3+34 x^2+44 x+22$
- $y^2=43 x^6+23 x^5+2 x^4+x^3+28 x^2+17 x+23$
- $y^2=44 x^6+21 x^5+18 x^4+41 x^3+19 x^2+x+40$
- $y^2=10 x^6+39 x^5+24 x^4+26 x^3+26 x^2+2 x+40$
- $y^2=35 x^6+11 x^5+15 x^4+20 x^3+39 x^2+45 x+15$
- $y^2=40 x^6+43 x^5+43 x^4+17 x^3+8 x^2+26 x+26$
- $y^2=22 x^6+33 x^5+16 x^4+34 x^3+21 x^2+21 x+14$
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 4.0.2490624.1. |
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.u_hg | $2$ | (not in LMFDB) |