Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x + 12 x^{2} - 28 x^{3} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.182041207691$, $\pm0.527071640754$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.131328.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $30$ | $2820$ | $116910$ | $5696400$ | $288519150$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $58$ | $340$ | $2374$ | $17164$ | $118906$ | $823708$ | $5761726$ | $40357060$ | $282469018$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+2x^5+5x^4+6x^3+4x^2+x+6$
- $y^2=5x^6+2x^5+6x^3+4x^2+3x$
- $y^2=4x^5+3x^4+4x^3+6x^2+5x+6$
- $y^2=3x^6+4x^5+4x^4+5x^3+x^2+6x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7}$.
Endomorphism algebra over $\F_{7}$| The endomorphism algebra of this simple isogeny class is 4.0.131328.1. |
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 |
|---|---|---|
| 2.7.e_m | $2$ | 2.49.i_s |