Invariants
Base field: | $\F_{229}$ |
Dimension: | $1$ |
L-polynomial: | $1 + 7 x + 229 x^{2}$ |
Frobenius angles: | $\pm0.574293476712$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-3}) \) |
Galois group: | $C_2$ |
Jacobians: | $7$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $237$ | $52851$ | $12004524$ | $2749996083$ | $629764851657$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $237$ | $52851$ | $12004524$ | $2749996083$ | $629764851657$ | $144215820874944$ | $33025421684948853$ | $7562821650526509123$ | $1731886157674507633596$ | $396601930090144520939451$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{229}$.
Endomorphism algebra over $\F_{229}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \). |
Base change
This is a primitive isogeny class.