Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x + 24 x^{2} - 44 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.327898829548$, $\pm0.471853187744$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.90368.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $98$ | $19012$ | $1898162$ | $213010448$ | $25827704098$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $154$ | $1424$ | $14550$ | $160368$ | $1771210$ | $19487000$ | $214347678$ | $2357974280$ | $25937834874$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=2 x^6+8 x^5+9 x^3+7 x^2+2 x$
- $y^2=8 x^5+10 x^4+8 x^3+4 x^2+2 x+1$
- $y^2=4 x^6+2 x^5+6 x^4+6 x^3+3 x^2+5 x+2$
- $y^2=8 x^6+5 x^5+4 x^4+x^3+8 x^2+3$
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 4.0.90368.2. |
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.e_y | $2$ | 2.121.bg_ry |