Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 5 x^{2} + 16 x^{4}$ |
| Frobenius angles: | $\pm0.357450520704$, $\pm0.642549479296$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{3}, \sqrt{-13})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 4 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $22$ | $484$ | $3982$ | $69696$ | $1048102$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $5$ | $27$ | $65$ | $271$ | $1025$ | $3867$ | $16385$ | $66463$ | $262145$ | $1047627$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x+a+1)y=(a+1)x^5+(a+1)x^3+(a+1)x^2+ax+a$
- $y^2+(x^2+x+a+1)y=ax^5+ax^3+ax^2+x+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2^{2}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{3}, \sqrt{-13})\). |
| The base change of $A$ to $\F_{2^{4}}$ is 1.16.f 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-39}) \)$)$ |
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 |
|---|---|---|
| 2.4.a_af | $4$ | 2.256.o_vp |
| 2.4.ad_h | $12$ | (not in LMFDB) |
| 2.4.d_h | $12$ | (not in LMFDB) |