Invariants
| Base field: | $\F_{163}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 4 x + 163 x^{2}$ |
| Frobenius angles: | $\pm0.550070136886$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-159}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $20$ |
| Isomorphism classes: | 20 |
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: | $1$ |
| Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $168$ | $26880$ | $4328856$ | $705868800$ | $115064097288$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $168$ | $26880$ | $4328856$ | $705868800$ | $115064097288$ | $18755374659840$ | $3057125142608376$ | $498311413884211200$ | $81224760551662338408$ | $13239635967017652998400$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which 0 are hyperelliptic):
- $y^2=x^3+120 x+120$
- $y^2=x^3+90 x+17$
- $y^2=x^3+149 x+149$
- $y^2=x^3+128 x+128$
- $y^2=x^3+50 x+100$
- $y^2=x^3+161 x+161$
- $y^2=x^3+40 x+80$
- $y^2=x^3+43 x+43$
- $y^2=x^3+38 x+38$
- $y^2=x^3+68 x+68$
- $y^2=x^3+54 x+108$
- $y^2=x^3+116 x+69$
- $y^2=x^3+18 x+36$
- $y^2=x^3+87 x+11$
- $y^2=x^3+90 x+90$
- $y^2=x^3+148 x+133$
- $y^2=x^3+95 x+27$
- $y^2=x^3+16 x+16$
- $y^2=x^3+147 x+131$
- $y^2=x^3+118 x+118$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{163}$.
Endomorphism algebra over $\F_{163}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-159}) \). |
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 |
|---|---|---|
| 1.163.ae | $2$ | (not in LMFDB) |