Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $1$ |
L-polynomial: | $1 + x + 8 x^{2}$ |
Frobenius angles: | $\pm0.556567041129$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-31}) \) |
Galois group: | $C_2$ |
Jacobians: | $3$ |
Isomorphism classes: | 3 |
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: | $1$ |
Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $10$ | $80$ | $490$ | $4000$ | $33050$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $80$ | $490$ | $4000$ | $33050$ | $262640$ | $2094410$ | $16776000$ | $134240890$ | $1073728400$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{3}}$.
Endomorphism algebra over $\F_{2^{3}}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-31}) \). |
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 |
---|---|---|
1.8.ab | $2$ | 1.64.p |