Invariants
| Base field: | $\F_{2}$ |
| Dimension: | $4$ |
| L-polynomial: | $1 + 3 x^{4} + 16 x^{8}$ |
| Frobenius angles: | $\pm0.155589323385$, $\pm0.344410676615$, $\pm0.655589323385$, $\pm0.844410676615$ |
| Angle rank: | $1$ (numerical) |
| Number field: | 8.0.2342560000.5 |
| Galois group: | $D_4\times C_2$ |
| Jacobians: | $5$ |
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: | $4$ |
| Slopes: | $[0, 0, 0, 0, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $20$ | $400$ | $3980$ | $160000$ | $1050500$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $5$ | $9$ | $29$ | $33$ | $65$ | $129$ | $349$ | $513$ | $1025$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which 2 are hyperelliptic):
- $y^2 + (x^4 + x + 1) y=x^9 + x^3 + x^2 + 1$
- $y^2 + (x^4 + x + 1) y=x^9 + x^3 + x^2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2}$| The endomorphism algebra of this simple isogeny class is 8.0.2342560000.5. |
| The base change of $A$ to $\F_{2^{4}}$ is 1.16.d 4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-55}) \)$)$ |
- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 2.4.a_d 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{5}, \sqrt{-11})\)$)$
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 4.2.a_a_a_ad | $8$ | (not in LMFDB) |