Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 5 x + 13 x^{2} + 35 x^{3} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.488412663036$, $\pm0.938355115093$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-58 +10 \sqrt{29}})\) |
| Galois group: | $C_4$ |
| Jacobians: | $1$ |
| Isomorphism classes: | 1 |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $103$ | $2369$ | $130501$ | $5375261$ | $284182768$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $13$ | $51$ | $379$ | $2235$ | $16908$ | $118047$ | $824389$ | $5760099$ | $40347343$ | $282518686$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):
- $y^2=5 x^6+2 x^5+6 x^4+x^3+4 x^2+4$
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 \(\Q(\sqrt{-58 +10 \sqrt{29}})\). |
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_n | $2$ | 2.49.b_adf |