Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 5 x + 20 x^{2} + 55 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.482125936541$, $\pm0.800479701061$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-118 +10 \sqrt{33}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $202$ | $16564$ | $1760632$ | $213741856$ | $25772846902$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $17$ | $137$ | $1322$ | $14601$ | $160027$ | $1776242$ | $19487737$ | $214323793$ | $2357978462$ | $25937375177$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=3 x^6+x^5+7 x^3+7 x^2+6 x+9$
- $y^2=9 x^5+3 x^4+7 x^3+7 x^2+2 x+9$
- $y^2=4 x^6+2 x^5+2 x^4+8 x^3+10 x^2+9 x+8$
- $y^2=10 x^6+4 x^5+2 x^4+6 x^3+10 x^2+3 x+3$
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 \(\Q(\sqrt{-118 +10 \sqrt{33}})\). |
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.af_u | $2$ | 2.121.p_do |