Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 3 x + 13 x^{2} + 33 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.409825870073$, $\pm0.761311735013$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-122 +18 \sqrt{5}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $15$ |
| Isomorphism classes: | 21 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $171$ | $16929$ | $1783701$ | $216369549$ | $25704421776$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $15$ | $139$ | $1341$ | $14779$ | $159600$ | $1771783$ | $19499775$ | $214349539$ | $2357980281$ | $25937005654$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 15 curves (of which all are hyperelliptic):
- $y^2=9 x^6+8 x^5+4 x^3+8 x+4$
- $y^2=5 x^6+6 x^5+2 x^4+3 x^3+10 x^2+10 x$
- $y^2=5 x^6+10 x^5+2 x^3+2 x^2+7 x+1$
- $y^2=8 x^6+2 x^5+9 x^4+9 x^3+8 x^2+6$
- $y^2=9 x^6+9 x^5+3 x^4+7 x^2+5 x+4$
- $y^2=x^6+3 x^5+10 x^4+9 x^3+3 x^2+6 x+1$
- $y^2=7 x^6+4 x^4+2 x^3+6 x^2+2 x+6$
- $y^2=10 x^6+8 x^5+9 x^4+2 x^3+10 x^2+4 x$
- $y^2=10 x^6+10 x^5+9 x^4+6 x^3+7 x^2+5 x+4$
- $y^2=5 x^5+3 x^4+x^3+7 x^2+6 x+9$
- $y^2=3 x^6+9 x^5+x^4+x^3+3 x^2+3$
- $y^2=9 x^6+5 x^5+2 x^4+7 x^3+4 x^2+2 x+1$
- $y^2=10 x^5+3 x^4+2 x^3+8 x^2+9 x+2$
- $y^2=9 x^6+9 x^5+5 x^4+8 x^3+x^2+5$
- $y^2=8 x^5+7 x^4+8 x^3+4 x^2+7 x+9$
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 \(\Q(\sqrt{-122 +18 \sqrt{5}})\). |
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.ad_n | $2$ | 2.121.r_if |