Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 2 x + 4 x^{2} + 22 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.330985876724$, $\pm0.799387471380$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-9 +2 \sqrt{19}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $12$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $150$ | $15300$ | $1839150$ | $219157200$ | $25748328750$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $14$ | $126$ | $1382$ | $14966$ | $159874$ | $1771038$ | $19479754$ | $214362526$ | $2358124142$ | $25937283726$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=4 x^6+10 x^5+3 x^4+9 x^3+x^2+2 x+3$
- $y^2=5 x^6+7 x^5+10 x^4+10 x^3+6 x^2+9 x+1$
- $y^2=3 x^6+6 x^5+9 x^3+x^2+4 x+3$
- $y^2=x^5+4 x^4+9 x^3+8 x^2+7 x+5$
- $y^2=5 x^6+8 x^5+5 x^4+8 x^3+5 x^2+5 x+1$
- $y^2=10 x^6+6 x^5+4 x^4+9 x^3+x^2+8$
- $y^2=2 x^6+2 x^5+5 x^4+2 x^3+4 x^2+x+6$
- $y^2=3 x^6+10 x^5+9 x^4+10 x^2+5 x+7$
- $y^2=7 x^6+9 x^5+9 x^4+8 x^3+9 x^2+7 x+4$
- $y^2=4 x^6+5 x^5+2 x^4+9 x^3+8 x^2+10 x$
- $y^2=4 x^6+8 x^5+4 x^4+6 x^3+7 x^2+5$
- $y^2=7 x^6+8 x^5+6 x^4+9 x^3+6 x^2+2 x+5$
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 \(\Q(\sqrt{-9 +2 \sqrt{19}})\). |
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_e | $2$ | 2.121.e_go |