Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 5 x + 13 x^{2} - 55 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.0837222189534$, $\pm0.567942489704$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.18605.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $5$ |
| Isomorphism classes: | 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $75$ | $14625$ | $1649925$ | $210322125$ | $26018130000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $123$ | $1237$ | $14363$ | $161552$ | $1772343$ | $19480727$ | $214377763$ | $2358108337$ | $25937533398$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=8x^6+8x^5+7x^3+8x+10$
- $y^2=10x^6+10x^5+3x^4+8x^3+7x^2+5x+3$
- $y^2=9x^6+7x^5+3x^4+8x^3+4x^2+2x+10$
- $y^2=2x^6+9x^5+7x^4+x^2+7$
- $y^2=8x^6+x^5+4x^4+7x^3+4x^2+6x+9$
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 4.0.18605.1. |
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.f_n | $2$ | 2.121.b_afj |