Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 2 x + 7 x^{2} )( 1 + 3 x + 7 x^{2} )$ |
| $1 + x + 8 x^{2} + 7 x^{3} + 49 x^{4}$ | |
| Frobenius angles: | $\pm0.376624142786$, $\pm0.691875465479$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $4$ |
| Isomorphism classes: | 26 |
| Cyclic group of points: | yes |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $66$ | $3300$ | $116424$ | $5940000$ | $278988006$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $9$ | $65$ | $342$ | $2473$ | $16599$ | $116570$ | $825897$ | $5769073$ | $40348314$ | $282482825$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=2 x^6+x^5+2 x^4+4 x^2+6 x$
- $y^2=x^6+5 x^5+6 x^4+6 x^3+2 x^2+2 x+3$
- $y^2=2 x^6+3 x^5+3 x^4+x^3+3 x^2+4 x+3$
- $y^2=3 x^6+2 x^5+x^4+3 x^3+4 x^2+x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7}$.
Endomorphism algebra over $\F_{7}$| The isogeny class factors as 1.7.ac $\times$ 1.7.d and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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.af_u | $2$ | 2.49.p_fs |
| 2.7.ab_i | $2$ | 2.49.p_fs |
| 2.7.f_u | $2$ | 2.49.p_fs |