Invariants
| Base field: | $\F_{2^{4}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 7 x + 16 x^{2} )( 1 - 4 x + 16 x^{2} )$ |
| $1 - 11 x + 60 x^{2} - 176 x^{3} + 256 x^{4}$ | |
| Frobenius angles: | $\pm0.160861246510$, $\pm0.333333333333$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $2$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $1$ |
| Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $130$ | $65520$ | $17280250$ | $4326547680$ | $1100192538250$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $256$ | $4218$ | $66016$ | $1049226$ | $16777168$ | $268449306$ | $4295114176$ | $68720086698$ | $1099511974576$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+xy=(a^2+a)x^5+(a^2+a)x^3+(a^3+1)x^2+x$
- $y^2+xy=(a^2+a+1)x^5+(a^2+a+1)x^3+(a^3+a^2+1)x^2+x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{12}}$.
Endomorphism algebra over $\F_{2^{4}}$| The isogeny class factors as 1.16.ah $\times$ 1.16.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ah $\times$ 1.4096.ey. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.