Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 3 x + 11 x^{2} )^{2}$ |
| $1 - 6 x + 31 x^{2} - 66 x^{3} + 121 x^{4}$ | |
| Frobenius angles: | $\pm0.350615407277$, $\pm0.350615407277$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $5$ |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $81$ | $18225$ | $1971216$ | $216531225$ | $25753509441$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $148$ | $1476$ | $14788$ | $159906$ | $1766518$ | $19484646$ | $214406788$ | $2358119196$ | $25937412148$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+x^5+5x^4+10x^3+5x^2+x+2$
- $y^2=7x^6+10x^5+8x^4+7x^3+8x^2+10x+7$
- $y^2=6x^6+x^5+x^4+x^3+9x^2+4x+7$
- $y^2=4x^6+3x^5+7x^4+9x^3+7x^2+3x+4$
- $y^2=2x^6+6x^5+2x^4+5x^3+2x^2+6x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11}$.
Endomorphism algebra over $\F_{11}$| The isogeny class factors as 1.11.ad 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-35}) \)$)$ |
Base change
This is a primitive isogeny class.