Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x + 5 x^{2} - 44 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.0393704996337$, $\pm0.627296167033$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-7})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $79$ | $13825$ | $1597696$ | $211370425$ | $25949509279$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $116$ | $1196$ | $14436$ | $161128$ | $1767638$ | $19478488$ | $214372036$ | $2357861876$ | $25937108276$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=8 x^6+7 x^4+3 x^3+4 x^2+9 x+6$
- $y^2=5 x^6+9 x^5+7 x^4+5 x^3+4 x^2+5 x+10$
- $y^2=2 x^6+5 x^5+3 x^4+5 x^3+7 x^2+2 x+8$
- $y^2=3 x^6+8 x^5+9 x^4+9 x^3+4 x+8$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{11^{3}}$.
Endomorphism algebra over $\F_{11}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-7})\). |
| The base change of $A$ to $\F_{11^{3}}$ is 1.1331.acq 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$ |
Base change
This is a primitive isogeny class.