Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - x + x^{2} - 7 x^{3} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.213996916938$, $\pm0.702220529456$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.553373.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $43$ | $2537$ | $111241$ | $6182669$ | $286607728$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $51$ | $325$ | $2571$ | $17052$ | $117543$ | $825307$ | $5760099$ | $40334437$ | $282478366$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):
- $y^2=4 x^5+2 x^4+x^2+6 x+4$
- $y^2=5 x^6+5 x^5+2 x^3+4 x^2+4 x+5$
- $y^2=3 x^6+5 x^5+4 x^4+6 x^2+2 x+4$
- $y^2=4 x^6+5 x^5+3 x^4+3 x^3+2$
- $y^2=5 x^6+x^5+3 x^4+6 x^3+4 x^2+6 x+3$
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 4.0.553373.1. |
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_b | $2$ | 2.49.b_dh |