Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 4 x + 16 x^{2} + 44 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.443936343938$, $\pm0.783888630716$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-30 +4 \sqrt{10}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $186$ | $16740$ | $1778346$ | $214874640$ | $25713399066$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $16$ | $138$ | $1336$ | $14678$ | $159656$ | $1774458$ | $19495856$ | $214334878$ | $2357946256$ | $25937081898$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=2 x^6+5 x^5+8 x^4+7 x^3+x^2+5 x$
- $y^2=3 x^6+6 x^5+6 x^4+2 x^3+4 x^2+x+5$
- $y^2=9 x^6+2 x^5+10 x^4+3 x^3+4 x^2+7 x+3$
- $y^2=x^6+2 x^4+6 x^3+2 x^2+6 x+6$
- $y^2=3 x^6+10 x^5+5 x^3+x^2+9 x$
- $y^2=6 x^6+6 x^5+4 x^4+8 x^3+9$
- $y^2=4 x^6+9 x^5+4 x^4+6 x^3+8 x^2+7 x+7$
- $y^2=9 x^6+4 x^4+5 x^3+10 x^2+4 x+10$
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 \(\Q(\sqrt{-30 +4 \sqrt{10}})\). |
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.ae_q | $2$ | 2.121.q_fq |