Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - x + 3 x^{2} - 7 x^{3} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.240284356173$, $\pm0.681339296947$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-66 -6 \sqrt{5}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 10 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $45$ | $2745$ | $112995$ | $6162525$ | $287298000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $55$ | $331$ | $2563$ | $17092$ | $117115$ | $823921$ | $5760883$ | $40330897$ | $282493150$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=3 x^6+6 x^5+3 x^4+2 x^3+4 x^2+6 x+2$
- $y^2=4 x^6+4 x^4+6 x^3+5 x+6$
- $y^2=4 x^6+5 x^5+x^4+3 x^3+4 x^2+5 x+3$
- $y^2=6 x^6+x^5+2 x^4+4 x^3+4 x^2+6 x+3$
- $y^2=3 x^6+2 x^5+5 x^4+6 x^3+5 x^2+6 x+3$
- $y^2=x^5+4 x^4+x^2+6 x+3$
- $y^2=5 x^6+6 x^5+5 x^4+6 x^3+x$
- $y^2=x^6+5 x^5+x^4+3 x^3+2 x$
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 \(\Q(\sqrt{-66 -6 \sqrt{5}})\). |
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.b_d | $2$ | 2.49.f_dp |