Invariants
| Base field: | $\F_{2^{6}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 15 x + 64 x^{2} )( 1 - 8 x + 64 x^{2} )$ |
| $1 - 23 x + 248 x^{2} - 1472 x^{3} + 4096 x^{4}$ | |
| Frobenius angles: | $\pm0.113134082257$, $\pm0.333333333333$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $6$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2850$ | $16644000$ | $68858168850$ | $281523306888000$ | $1152900736926299250$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $42$ | $4064$ | $262674$ | $16780096$ | $1073722482$ | $68719231712$ | $4398047743698$ | $281475025561216$ | $18014399045913906$ | $1152921507647841824$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+xy=(a^3+1)x^5+a^2x^3+(a^3+a+1)x^2+x$
- $y^2+xy=(a^5+a^3+a^2+a)x^5+(a^5+a^4+a)x^3+(a^4+a^3+a^2+1)x^2+x$
- $y^2+xy=(a^4+a^3+a)x^5+a^4x^3+(a^5+a^4+a^3+a^2+a+1)x^2+x$
- $y^2+xy=(a^5+a^4+a^3+a^2+a)x^5+ax^3+(a^4+a^3+1)x^2+x$
- $y^2+xy=(a^5+a^3+a)x^5+(a^4+a+1)x^3+(a^5+a^4+a^3+a^2+1)x^2+x$
- $y^2+xy=(a^5+a^3)x^5+(a^5+a^4+a^2+a+1)x^3+(a^3+a)x^2+x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{18}}$.
Endomorphism algebra over $\F_{2^{6}}$| The isogeny class factors as 1.64.ap $\times$ 1.64.ai 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 $\F_{2^{18}}$ is 1.262144.atb $\times$ 1.262144.bnk. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.