Invariants
Base field: | $\F_{241}$ |
Dimension: | $1$ |
L-polynomial: | $1 - 31 x + 241 x^{2}$ |
Frobenius angles: | $\pm0.0177663311058$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-3}) \) |
Galois group: | $C_2$ |
Jacobians: | $1$ |
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: | $1$ |
Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $211$ | $57603$ | $13990144$ | $3373289283$ | $812988283651$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $211$ | $57603$ | $13990144$ | $3373289283$ | $812988283651$ | $195930567705600$ | $47219272787201971$ | $11379844832476032003$ | $2742542606001488007424$ | $660952768067103056839203$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{241}$.
Endomorphism algebra over $\F_{241}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \). |
Base change
This is a primitive isogeny class.