Invariants
| Base field: | $\F_{2^{4}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 5 x + 16 x^{2} )^{2}$ |
| $1 - 10 x + 57 x^{2} - 160 x^{3} + 256 x^{4}$ | |
| Frobenius angles: | $\pm0.285098958592$, $\pm0.285098958592$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $6$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $144$ | $69696$ | $17740944$ | $4356000000$ | $1100510098704$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $271$ | $4327$ | $66463$ | $1049527$ | $16767151$ | $268369927$ | $4294800703$ | $68719692247$ | $1099515370831$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x)y=(a^2+a+1)x^5+(a^2+a+1)x^3+(a^2+a+1)x^2+(a^2+a+1)x$
- $y^2+(x^2+x)y=(a^2+a)x^5+(a^2+a)x^3+(a^2+a)x^2+(a^2+a)x$
- $y^2+(x^2+x)y=(a^2+1)x^5+(a^2+1)x^3+(a+1)x^2+(a+1)x$
- $y^2+(x^2+x)y=ax^5+ax^3+(a^2+1)x^2+(a^2+1)x$
- $y^2+(x^2+x)y=a^2x^5+a^2x^3+ax^2+ax$
- $y^2+(x^2+x)y=(a+1)x^5+(a+1)x^3+a^2x^2+a^2x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2^{4}}$| The isogeny class factors as 1.16.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-39}) \)$)$ |
Base change
This is a primitive isogeny class.