Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 2 x + 20 x^{2} + 22 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.464799360054$, $\pm0.635124972798$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.228672.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 6 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $166$ | $19588$ | $1711294$ | $211628752$ | $25979654206$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $14$ | $158$ | $1286$ | $14454$ | $161314$ | $1771454$ | $19491850$ | $214368670$ | $2357805230$ | $25937423918$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=7 x^6+x^5+6 x^4+9 x^2+5 x+5$
- $y^2=3 x^6+7 x^5+2 x^4+8 x^3+x^2+9 x+9$
- $y^2=7 x^6+x^5+6 x^4+10 x^3+9 x^2+x+10$
- $y^2=8 x^5+x^4+4 x^3+7 x^2+x$
- $y^2=7 x^6+4 x^5+8 x^4+9 x^3+6 x^2+4$
- $y^2=9 x^6+8 x^5+3 x^4+10 x^3+x+4$
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 4.0.228672.1. |
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.ac_u | $2$ | 2.121.bk_vi |