Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 3 x + 7 x^{2} - 21 x^{3} + 49 x^{4}$ |
Frobenius angles: | $\pm0.171557865674$, $\pm0.594085680913$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.28749.1 |
Galois group: | $D_{4}$ |
Jacobians: | $6$ |
Isomorphism classes: | 6 |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $33$ | $2673$ | $108999$ | $5808429$ | $290838768$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $55$ | $317$ | $2419$ | $17300$ | $118195$ | $823415$ | $5770099$ | $40357739$ | $282421150$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+5x^5+x^4+6x^3+6x^2+x+2$
- $y^2=2x^6+x^5+x^4+5x^3+2x^2+3$
- $y^2=5x^5+5x^4+5x^3+3x^2+3x+6$
- $y^2=6x^6+6x^5+3x^4+2x^3+6$
- $y^2=6x^6+x^5+2x^3+4x^2+6x+6$
- $y^2=x^6+5x^5+3x^4+3x^3+5x^2+4x+3$
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 4.0.28749.1. |
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.d_h | $2$ | 2.49.f_v |