Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + x + 6 x^{2} + 7 x^{3} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.352022857153$, $\pm0.719949304674$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-78 +2 \sqrt{33}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $64$ | $3072$ | $118528$ | $6057984$ | $278002624$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $9$ | $61$ | $348$ | $2521$ | $16539$ | $116638$ | $825141$ | $5765329$ | $40363764$ | $282500341$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=3 x^6+2 x^5+4 x^4+3 x^3+2$
- $y^2=3 x^6+2 x^5+x^4+4 x^3+6 x^2+2 x+1$
- $y^2=6 x^6+2 x^5+4 x^4+4 x^3+6 x^2+6 x+1$
- $y^2=6 x^6+x^5+4 x^4+2 x^2+3 x+6$
- $y^2=5 x^5+2 x^3+4 x^2+6 x+4$
- $y^2=4 x^5+2 x^4+4 x^3+x^2+5 x+1$
- $y^2=3 x^5+5 x^4+3 x+1$
- $y^2=3 x^6+5 x^5+x^4+x^3+4 x^2+4 x+2$
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 \(\Q(\sqrt{-78 +2 \sqrt{33}})\). |
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.ab_g | $2$ | 2.49.l_eq |