Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + x + 5 x^{2} + 4 x^{3} + 16 x^{4}$ |
| Frobenius angles: | $\pm0.394402107922$, $\pm0.695268008842$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-18 +2 \sqrt{13}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $3$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $27$ | $459$ | $3888$ | $70227$ | $989847$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $26$ | $63$ | $274$ | $966$ | $3935$ | $16806$ | $65890$ | $261279$ | $1048586$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):
- $y^2+(x^3+x+1) y=x^3+x+1$
- $y^2+(x^3+a) y=a x^4+(a+1) x$
- $y^2+(x^3+a+1) y=(a+1) x^4+a x$
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 \(\Q(\sqrt{-18 +2 \sqrt{13}})\). |
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.ab_b |
| $\F_{2}$ | 2.2.b_b |
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.4.ab_f | $2$ | 2.16.j_bx |