Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 7 x + 25 x^{2} - 49 x^{3} + 49 x^{4}$ |
Frobenius angles: | $\pm0.162349854003$, $\pm0.351370772325$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.3725.1 |
Galois group: | $D_{4}$ |
Jacobians: | $1$ |
Isomorphism classes: | 1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $19$ | $2489$ | $130321$ | $5946221$ | $282939184$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $1$ | $51$ | $379$ | $2475$ | $16836$ | $117687$ | $824965$ | $5771619$ | $40366243$ | $282461086$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+3x^5+3x^4+4x^3+4x^2+4x+3$
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 4.0.3725.1. |
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.h_z | $2$ | 2.49.b_bl |