Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 6 x + 29 x^{2} + 66 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.576841329034$, $\pm0.731767881221$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-33 -6 \sqrt{2}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 2 |
| Cyclic group of points: | yes |
This isogeny class is simple and 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $223$ | $17617$ | $1633252$ | $215790633$ | $25996403623$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $144$ | $1224$ | $14740$ | $161418$ | $1770990$ | $19486590$ | $214342948$ | $2358053640$ | $25937358624$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):
- $y^2=4 x^6+6 x^4+4 x^3+10 x^2+2 x+5$
- $y^2=6 x^6+10 x^5+6 x^4+7 x^3+2 x^2+2 x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11}$.
Endomorphism algebra over $\F_{11}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-33 -6 \sqrt{2}})\). |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.11.ag_bd | $2$ | 2.121.w_lf |