Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + x + 19 x^{2} + 11 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.437074627469$, $\pm0.612852731135$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-162 +2 \sqrt{13}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $5$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $153$ | $19737$ | $1740987$ | $211363533$ | $25955901648$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $13$ | $159$ | $1309$ | $14435$ | $161168$ | $1771155$ | $19490435$ | $214387123$ | $2357847895$ | $25937001174$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):
- $y^2=4 x^6+2 x^5+x^4+10 x^3+10 x^2+4 x+5$
- $y^2=5 x^6+3 x^5+3 x^4+5 x^3+5 x^2+2 x+5$
- $y^2=10 x^6+5 x^5+7 x^4+6 x^3+2 x^2+x+4$
- $y^2=7 x^6+4 x^5+x^4+x^3+8 x^2+6 x+7$
- $y^2=5 x^6+3 x^5+5 x^4+4 x^3+x^2+10 x+5$
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 \(\Q(\sqrt{-162 +2 \sqrt{13}})\). |
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.ab_t | $2$ | 2.121.bl_wj |