Invariants
| Base field: | $\F_{127}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 - 2 x + 127 x^{2}$ |
| Frobenius angles: | $\pm0.471717365588$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-14}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $12$ |
| Isomorphism classes: | 12 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $126$ | $16380$ | $2049138$ | $260114400$ | $33038213166$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $126$ | $16380$ | $2049138$ | $260114400$ | $33038213166$ | $4195876442940$ | $532875887064738$ | $67675233846729600$ | $8594754744404616606$ | $1091533853115058707900$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which 0 are hyperelliptic):
- $y^2=x^3+116 x+94$
- $y^2=x^3+52 x+29$
- $y^2=x^3+114 x+88$
- $y^2=x^3+11 x+11$
- $y^2=x^3+106 x+106$
- $y^2=x^3+51 x+51$
- $y^2=x^3+85 x+85$
- $y^2=x^3+19 x+19$
- $y^2=x^3+59 x+59$
- $y^2=x^3+38 x+38$
- $y^2=x^3+x+1$
- $y^2=x^3+50 x+50$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{127}$.
Endomorphism algebra over $\F_{127}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-14}) \). |
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 |
|---|---|---|
| 1.127.c | $2$ | (not in LMFDB) |