Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 3 x + 11 x^{2} - 21 x^{3} + 49 x^{4}$ |
Frobenius angles: | $\pm0.245749335339$, $\pm0.547779066685$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.164493.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $37$ | $3145$ | $121027$ | $5802525$ | $288795952$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $63$ | $353$ | $2419$ | $17180$ | $118131$ | $820811$ | $5758291$ | $40356011$ | $282473118$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=4x^6+2x^5+3x^4+4x^3+6x^2+2x+5$
- $y^2=5x^6+4x^5+4x^4+4x^3+3x^2+5x+4$
- $y^2=x^6+3x^5+2x^4+4x^3+4x^2+4x+2$
- $y^2=3x^6+x^5+3x^4+3x^3+3x^2+6x$
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 4.0.164493.1. |
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_l | $2$ | 2.49.n_dp |