Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 13 x^{2} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.149384592723$, $\pm0.850615407277$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{35})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
This isogeny class is simple but not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $109$ | $11881$ | $1774084$ | $216531225$ | $25937430829$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $96$ | $1332$ | $14788$ | $161052$ | $1776606$ | $19487172$ | $214406788$ | $2357947692$ | $25937437056$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=x^6+5 x^5+3 x^4+10 x^3+6 x^2+3 x+2$
- $y^2=7 x^6+3 x^4+4 x^3+7 x^2+3 x+4$
- $y^2=3 x^6+6 x^4+8 x^3+3 x^2+6 x+8$
- $y^2=2 x^6+4 x^5+2 x^4+2 x+6$
- $y^2=7 x^6+x^5+3 x^4+10 x^3+10 x^2+4 x+10$
- $y^2=3 x^6+2 x^5+6 x^4+9 x^3+9 x^2+8 x+9$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11^{2}}$.
Endomorphism algebra over $\F_{11}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{35})\). |
| The base change of $A$ to $\F_{11^{2}}$ is 1.121.an 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-35}) \)$)$ |
Base change
This is a primitive isogeny class.