Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 2 x + 10 x^{2} - 14 x^{3} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.290542551506$, $\pm0.575048949275$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.11600.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $44$ | $3344$ | $120956$ | $5831936$ | $287190684$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $66$ | $354$ | $2430$ | $17086$ | $117282$ | $819930$ | $5763774$ | $40368918$ | $282483586$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=6 x^6+5 x^4+2 x^3+4 x^2+2 x+1$
- $y^2=6 x^6+5 x^5+x^4+3 x^3+3 x^2+6$
- $y^2=x^6+x^5+6 x^4+5 x^3+2 x$
- $y^2=6 x^6+5 x^5+x^4+x^3+4 x^2+x+5$
- $y^2=4 x^6+x^4+4 x^3+3 x^2+2 x+5$
- $y^2=4 x^5+2 x^4+x^3+4 x^2+x+2$
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 4.0.11600.1. |
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.c_k | $2$ | 2.49.q_fm |