Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 3 x - 2 x^{2} + 33 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.316051259389$, $\pm0.982717926056$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-35})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 4 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $156$ | $13104$ | $1971216$ | $213280704$ | $26029874676$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $15$ | $109$ | $1476$ | $14569$ | $161625$ | $1766518$ | $19488435$ | $214334929$ | $2358119196$ | $25937430829$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):
- $y^2=5 x^6+7 x^5+9 x^4+5 x^2+x+5$
- $y^2=5 x^6+5 x^4+5 x^3+3 x^2+8 x+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11^{3}}$.
Endomorphism algebra over $\F_{11}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-35})\). |
| The base change of $A$ to $\F_{11^{3}}$ is 1.1331.cu 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-35}) \)$)$ |
Base change
This is a primitive isogeny class.