Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 4 x + 5 x^{2} + 44 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.372703832967$, $\pm0.960629500366$ |
| 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: | no |
| Non-cyclic primes: | $5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $175$ | $13825$ | $1960000$ | $211370425$ | $25925029375$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $16$ | $116$ | $1468$ | $14436$ | $160976$ | $1767638$ | $19495856$ | $214372036$ | $2358033508$ | $25937108276$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=4 x^6+9 x^4+7 x^3+2 x^2+10 x+3$
- $y^2=8 x^6+10 x^5+9 x^4+8 x^3+2 x^2+8 x+5$
- $y^2=4 x^6+10 x^5+6 x^4+10 x^3+3 x^2+4 x+5$
- $y^2=6 x^6+5 x^5+7 x^4+7 x^3+8 x+5$
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.cq 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$ |
Base change
This is a primitive isogeny class.