Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 4 x + 13 x^{2} - 28 x^{3} + 49 x^{4}$ |
Frobenius angles: | $\pm0.204545263622$, $\pm0.514205363720$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.7025.1 |
Galois group: | $D_{4}$ |
Jacobians: | $6$ |
Isomorphism classes: | 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $31$ | $2945$ | $121024$ | $5728025$ | $287835031$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $60$ | $352$ | $2388$ | $17124$ | $118830$ | $823372$ | $5758308$ | $40347424$ | $282472300$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+6x^5+5x^3+6x+6$
- $y^2=3x^6+4x^5+2x^4+4x^3+2x^2+4x+4$
- $y^2=3x^6+4x^5+x^4+3x^3+x^2+4x+2$
- $y^2=6x^6+3x^5+6x^4+x^3+3x^2+2x+3$
- $y^2=6x^6+3x^5+x^4+4x^3+2x^2+x+5$
- $y^2=6x^6+6x^5+3x^4+5x^3+3x^2+6x+1$
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 4.0.7025.1. |
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.e_n | $2$ | 2.49.k_br |