Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 2 x + x^{2} + 8 x^{3} + 16 x^{4}$ |
| Frobenius angles: | $\pm0.348885720110$, $\pm0.906432687576$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.1088.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $3$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is simple and geometrically simple, not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $28$ | $224$ | $6076$ | $65408$ | $1011388$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $15$ | $91$ | $255$ | $987$ | $3999$ | $16219$ | $66303$ | $262171$ | $1050655$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):
- $y^2+(x^2+x) y=x^5+(a+1) x^4+a x$
- $y^2+(x^2+x+a+1) y=x^5+(a+1) x^3+x+1$
- $y^2+(x^2+x+a) y=x^5+a x^3+x+1$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2^{2}}$| The endomorphism algebra of this simple isogeny class is 4.0.1088.2. |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.
| Subfield | Primitive Model |
| $\F_{2}$ | 2.2.ac_d |
| $\F_{2}$ | 2.2.c_d |
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.4.ac_b | $2$ | 2.16.ac_b |