Invariants
Base field: | $\F_{227}$ |
Dimension: | $1$ |
L-polynomial: | $1 + 227 x^{2}$ |
Frobenius angles: | $\pm0.5$ |
Angle rank: | $0$ (numerical) |
Number field: | \(\Q(\sqrt{-227}) \) |
Galois group: | $C_2$ |
Jacobians: | $20$ |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
$p$-rank: | $0$ |
Slopes: | $[1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $228$ | $51984$ | $11697084$ | $2655134784$ | $602738989908$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $228$ | $51984$ | $11697084$ | $2655134784$ | $602738989908$ | $136821774103056$ | $31058537410917804$ | $7050287986967865600$ | $1600415374247183470788$ | $363294289955316125848464$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 20 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{227^{2}}$.
Endomorphism algebra over $\F_{227}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-227}) \). |
The base change of $A$ to $\F_{227^{2}}$ is the simple isogeny class 1.51529.rm and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $227$ and $\infty$. |
Base change
This is a primitive isogeny class.
Twists
This isogeny class has no twists.