Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 9 x + 38 x^{2} + 99 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.619823774408$, $\pm0.953157107742$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{17})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $268$ | $13936$ | $1769872$ | $211214016$ | $26157070948$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $21$ | $117$ | $1332$ | $14425$ | $162411$ | $1768182$ | $19487433$ | $214376689$ | $2357947692$ | $25937131077$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):
- $y^2=9 x^6+4 x^5+10 x^4+2 x^3+4 x^2+6 x+9$
- $y^2=4 x^6+x^5+4 x^4+x^2+7 x+8$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11^{6}}$.
Endomorphism algebra over $\F_{11}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{17})\). |
| The base change of $A$ to $\F_{11^{6}}$ is 1.1771561.acna 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$ |
- Endomorphism algebra over $\F_{11^{2}}$
The base change of $A$ to $\F_{11^{2}}$ is the simple isogeny class 2.121.af_ads and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{17})\). - Endomorphism algebra over $\F_{11^{3}}$
The base change of $A$ to $\F_{11^{3}}$ is the simple isogeny class 2.1331.a_acna and its endomorphism algebra is \(\Q(\sqrt{-3}, \sqrt{17})\).
Base change
This is a primitive isogeny class.