Invariants
| Base field: | $\F_{11}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 5 x + 14 x^{2} - 55 x^{3} + 121 x^{4}$ |
| Frobenius angles: | $\pm0.105104110453$, $\pm0.561562556214$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-19})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $7$ |
| Isomorphism classes: | 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $76$ | $14896$ | $1669264$ | $210986944$ | $26054551876$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $125$ | $1252$ | $14409$ | $161777$ | $1773686$ | $19484507$ | $214383889$ | $2358139132$ | $25937628125$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=4x^6+8x^5+4x^4+9x^3+6x+10$
- $y^2=2x^6+7x^5+x^4+8x^3+3x^2+10x+6$
- $y^2=10x^6+8x^5+5x^4+5x^3+3x^2+7x+9$
- $y^2=4x^6+9x^5+10x^3+8x^2+2x+2$
- $y^2=8x^6+5x^5+8x^4+x^3+10x^2+4x+7$
- $y^2=10x^6+2x^5+10x^4+4x^3+9x^2+4x$
- $y^2=8x^6+8x^5+10x^4+5x^3+9x^2+4x+10$
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 \(\Q(\sqrt{-3}, \sqrt{-19})\). |
| The base change of $A$ to $\F_{11^{3}}$ is 1.1331.abo 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$ |
Base change
This is a primitive isogeny class.