Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 8 x + 33 x^{2} - 88 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.110710227191$, $\pm0.414323386517$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.5225.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $1$ |
| Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $59$ | $14809$ | $1793600$ | $212079689$ | $25833279859$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $124$ | $1348$ | $14484$ | $160404$ | $1772758$ | $19502284$ | $214402404$ | $2357980828$ | $25937438604$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=10x^6+7x^5+3x^4+6x^3+4x^2+7x+6$
- $y^2=8x^6+7x^5+5x^4+4x^3+7x^2+9x+10$
- $y^2=10x^6+7x^4+3x^3+9x^2+x+10$
- $y^2=8x^6+4x^5+2x^4+3x^3+10x^2+x+7$
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 4.0.5225.1. |
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.i_bh | $2$ | 2.121.c_acz |