Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 3 x + 7 x^{2} )( 1 + 2 x + 7 x^{2} )$ |
| $1 - x + 8 x^{2} - 7 x^{3} + 49 x^{4}$ | |
| Frobenius angles: | $\pm0.308124534521$, $\pm0.623375857214$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $4$ |
| Isomorphism classes: | 26 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $50$ | $3300$ | $117800$ | $5940000$ | $286013750$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $65$ | $346$ | $2473$ | $17017$ | $116570$ | $821191$ | $5769073$ | $40358902$ | $282482825$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=6 x^6+3 x^5+6 x^4+5 x^2+4 x$
- $y^2=5 x^6+4 x^5+2 x^4+2 x^3+3 x^2+3 x+1$
- $y^2=3 x^6+x^5+x^4+5 x^3+x^2+6 x+1$
- $y^2=2 x^6+6 x^5+3 x^4+2 x^3+5 x^2+3 x+6$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7}$.
Endomorphism algebra over $\F_{7}$| The isogeny class factors as 1.7.ad $\times$ 1.7.c 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.b_i | $2$ | 2.49.p_fs |
| 2.7.f_u | $2$ | 2.49.p_fs |