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: | $\pm0.649384592723$, $\pm0.649384592723$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $5$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3, 5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $225$ | $18225$ | $1587600$ | $216531225$ | $26122640625$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $148$ | $1188$ | $14788$ | $162198$ | $1766518$ | $19489698$ | $214406788$ | $2357776188$ | $25937412148$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):
- $y^2=4 x^6+2 x^5+10 x^4+9 x^3+10 x^2+2 x+4$
- $y^2=9 x^6+5 x^5+4 x^4+9 x^3+4 x^2+5 x+9$
- $y^2=3 x^6+6 x^5+6 x^4+6 x^3+10 x^2+2 x+9$
- $y^2=8 x^6+6 x^5+3 x^4+7 x^3+3 x^2+6 x+8$
- $y^2=x^6+3 x^5+x^4+8 x^3+x^2+3 x+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11}$.
Endomorphism algebra over $\F_{11}$| The isogeny class factors as 1.11.d 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-35}) \)$)$ |
Base change
This is a primitive isogeny class.