Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $1 - x - x^{2} - 7 x^{3} + 49 x^{4}$ |
Frobenius angles: | $\pm0.186914372711$, $\pm0.722521724645$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.524661.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
Isomorphism classes: | 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $41$ | $2337$ | $109511$ | $6162669$ | $285242576$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $7$ | $47$ | $319$ | $2563$ | $16972$ | $117947$ | $826357$ | $5761171$ | $40346833$ | $282469982$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+5x^5+5x^4+5x^3+4x^2+2x+6$
- $y^2=x^6+x^5+3x^2+2x+6$
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 4.0.524661.1. |
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_ab | $2$ | 2.49.ad_dh |