Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 5 x^{2} + 16 x^{4}$ |
| Frobenius angles: | $\pm0.142549479296$, $\pm0.857450520704$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{13})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $2$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $12$ | $144$ | $4212$ | $69696$ | $1049052$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $5$ | $7$ | $65$ | $271$ | $1025$ | $4327$ | $16385$ | $66463$ | $262145$ | $1049527$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):
- $y^2+(x^2+x) y=(a+1) x^5+(a+1) x^3+(a+1) x^2+(a+1) x$
- $y^2+(x^2+x) y=a x^5+a x^3+a x^2+a x$
where $a$ is a root of the Conway polynomial.
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.af 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.ad_h | $3$ | 2.64.a_el |
| 2.4.d_h | $3$ | 2.64.a_el |
| 2.4.a_f | $4$ | 2.256.o_vp |