Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x + 16 x^{2} - 44 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.216111369284$, $\pm0.556063656062$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.1184000.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 16 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $90$ | $16740$ | $1767690$ | $214874640$ | $26163056250$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $138$ | $1328$ | $14678$ | $162448$ | $1774458$ | $19478488$ | $214334878$ | $2357949128$ | $25937081898$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=x^6+8 x^5+4 x^4+9 x^3+6 x^2+8 x$
- $y^2=7 x^6+3 x^5+3 x^4+x^3+2 x^2+6 x+8$
- $y^2=7 x^6+4 x^5+9 x^4+6 x^3+8 x^2+3 x+6$
- $y^2=6 x^6+x^4+3 x^3+x^2+3 x+3$
- $y^2=6 x^6+9 x^5+10 x^3+2 x^2+7 x$
- $y^2=x^6+x^5+8 x^4+5 x^3+7$
- $y^2=8 x^6+7 x^5+8 x^4+x^3+5 x^2+3 x+3$
- $y^2=10 x^6+2 x^4+8 x^3+5 x^2+2 x+5$
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 4.0.1184000.2. |
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_q | $2$ | 2.121.q_fq |