Invariants
Base field: | $\F_{331}$ |
Dimension: | $1$ |
L-polynomial: | $1 + 331 x^{2}$ |
Frobenius angles: | $\pm0.5$ |
Angle rank: | $0$ (numerical) |
Number field: | \(\Q(\sqrt{-331}) \) |
Galois group: | $C_2$ |
Jacobians: | $12$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $332$ | $110224$ | $36264692$ | $12003393600$ | $3973195810652$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $332$ | $110224$ | $36264692$ | $12003393600$ | $3973195810652$ | $1315127885854864$ | $435307306210734212$ | $144086718331745798400$ | $47692703775754250891372$ | $15786284949782603436665104$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{331^{2}}$.
Endomorphism algebra over $\F_{331}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-331}) \). |
The base change of $A$ to $\F_{331^{2}}$ is the simple isogeny class 1.109561.zm and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $331$ and $\infty$. |
Base change
This is a primitive isogeny class.
Twists
This isogeny class has no twists.