Invariants
Base field: | $\F_{11^{2}}$ |
Dimension: | $1$ |
L-polynomial: | $( 1 - 11 x )^{2}$ |
$1 - 22 x + 121 x^{2}$ | |
Frobenius angles: | $0$, $0$ |
Angle rank: | $0$ (numerical) |
Number field: | \(\Q\) |
Galois group: | Trivial |
Jacobians: | $2$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $100$ | $14400$ | $1768900$ | $214329600$ | $25937102500$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $100$ | $14400$ | $1768900$ | $214329600$ | $25937102500$ | $3138424833600$ | $379749794608900$ | $45949729434854400$ | $5559917308776336100$ | $672749994880685160000$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11^{2}}$.
Endomorphism algebra over $\F_{11^{2}}$The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q\) ramified at $11$ and $\infty$. |
Base change
This is a primitive isogeny class.