Invariants
| Base field: | $\F_{2^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 3 x + 4 x^{2} )( 1 + x + 4 x^{2} )$ |
| $1 - 2 x + 5 x^{2} - 8 x^{3} + 16 x^{4}$ | |
| Frobenius angles: | $\pm0.230053456163$, $\pm0.580430623255$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $4$ |
| Isomorphism classes: | 20 |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $12$ | $384$ | $3996$ | $69120$ | $1175052$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $23$ | $63$ | $271$ | $1143$ | $4151$ | $16047$ | $65311$ | $261927$ | $1045703$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x+a+1)y=x^5+ax^4+ax^3+ax^2+(a+1)x+1$
- $y^2+(x^2+x+a)y=x^5+ax^4+(a+1)x^3+ax^2+ax+a$
- $y^2+(x^2+x)y=x^5+(a+1)x^4+x^3+(a+1)x$
- $y^2+(x^2+x)y=x^5+(a+1)x^4+x^3+x^2+ax$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2^{2}}$| The isogeny class factors as 1.4.ad $\times$ 1.4.b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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.ae_l | $2$ | 2.16.g_z |
| 2.4.c_f | $2$ | 2.16.g_z |
| 2.4.e_l | $2$ | 2.16.g_z |