Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + x - 10 x^{2} + 11 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.214837341119$, $\pm0.881504007785$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-43})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $3$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $124$ | $12400$ | $1860496$ | $217297600$ | $26026306804$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $13$ | $101$ | $1396$ | $14841$ | $161603$ | $1774838$ | $19479473$ | $214369201$ | $2357757676$ | $25937406101$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):
- $y^2=2 x^6+4 x^5+10 x^4+5 x^3+5 x+5$
- $y^2=9 x^6+9 x^5+10 x^4+6 x^3+2 x^2+5 x+10$
- $y^2=3 x^6+8 x^5+4 x^4+6 x^3+2 x^2+5 x+5$
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 \(\Q(\sqrt{-3}, \sqrt{-43})\). |
| The base change of $A$ to $\F_{11^{3}}$ is 1.1331.bg 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$ |
Base change
This is a primitive isogeny class.