Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 3 x + 9 x^{2} - 21 x^{3} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.208871173561$, $\pm0.572361342354$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.232957.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $5$ |
| Isomorphism classes: | 5 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $35$ | $2905$ | $114905$ | $5824525$ | $290838800$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $5$ | $59$ | $335$ | $2427$ | $17300$ | $118271$ | $821945$ | $5763523$ | $40353095$ | $282421214$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=4x^6+2x^4+2x^2+5x$
- $y^2=5x^6+2x^4+x^2+6x+5$
- $y^2=3x^6+4x^5+5x^4+3x^3+x+4$
- $y^2=5x^6+5x^5+2x^3+6x^2+2x+4$
- $y^2=3x^6+2x^5+6x^4+4x^3+5x^2+5x+3$
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 4.0.232957.1. |
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.d_j | $2$ | 2.49.j_cb |