Invariants
| Base field: | $\F_{47}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 20 x + 183 x^{2} - 940 x^{3} + 2209 x^{4}$ |
| Frobenius angles: | $\pm0.0765593021944$, $\pm0.337932522806$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.2960144.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 6 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1433$ | $4804849$ | $10795723316$ | $23808646620521$ | $52592734154109193$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $28$ | $2176$ | $103984$ | $4879140$ | $229317108$ | $10778982022$ | $506622697084$ | $23811295860804$ | $1119130576883728$ | $52599132745424736$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=40 x^6+31 x^5+23 x^4+11 x^3+40 x^2+14 x+22$
- $y^2=39 x^6+6 x^5+24 x^4+6 x^3+45 x^2+4 x+40$
- $y^2=24 x^6+37 x^5+31 x^4+6 x^3+4 x^2+29 x+43$
- $y^2=31 x^6+7 x^5+41 x^4+29 x^3+43 x^2+42 x+23$
- $y^2=26 x^6+12 x^5+38 x^4+4 x^3+23 x+6$
- $y^2=39 x^6+x^5+21 x^4+33 x^3+28 x^2+12 x+18$
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 4.0.2960144.1. |
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_hb | $2$ | (not in LMFDB) |