Invariants
Base field: | $\F_{2^{4}}$ |
Dimension: | $1$ |
L-polynomial: | $1 - x + 16 x^{2}$ |
Frobenius angles: | $\pm0.460106912325$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-7}) \) |
Galois group: | $C_2$ |
Jacobians: | $5$ |
Isomorphism classes: | 5 |
This isogeny class is simple and geometrically simple, not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $16$ | $288$ | $4144$ | $65088$ | $1047376$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $16$ | $288$ | $4144$ | $65088$ | $1047376$ | $16783200$ | $268460656$ | $4294896768$ | $68719003024$ | $1099512282528$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2^{4}}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-7}) \). |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{4}}$.
Subfield | Primitive Model |
$\F_{2}$ | 1.2.ab |
$\F_{2}$ | 1.2.b |
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
1.16.b | $2$ | 1.256.bf |