Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 5 x + 15 x^{2} - 35 x^{3} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.139519842760$, $\pm0.487441680688$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.62181.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $25$ | $2625$ | $116575$ | $5578125$ | $284182000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $55$ | $339$ | $2323$ | $16908$ | $118915$ | $825849$ | $5764723$ | $40358013$ | $282517150$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=5x^6+4x^5+3x^4+2x^3+3x^2+2x$
- $y^2=3x^6+x^5+3x^4+x^3+5x^2+6x$
- $y^2=5x^6+2x^4+4x^3+4x^2+2x$
- $y^2=5x^6+x^5+6x^4+3x^3+2x^2+5x+5$
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 4.0.62181.1. |
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.f_p | $2$ | 2.49.f_abb |