Invariants
| Base field: | $\F_{3^{4}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 9 x )^{2}( 1 - 14 x + 81 x^{2} )$ |
| $1 - 32 x + 414 x^{2} - 2592 x^{3} + 6561 x^{4}$ | |
| Frobenius angles: | $0$, $0$, $\pm0.216346895939$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4352$ | $41779200$ | $282004486400$ | $1852970355916800$ | $12157651951711430912$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $50$ | $6366$ | $530642$ | $43045566$ | $3486780530$ | $282429103518$ | $22876782455570$ | $1853020045666686$ | $150094633757983922$ | $12157665446009349726$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a^3+a+2)x^6+(2a^3+a^2+2a+2)x^5+(a^3+2a^2+a+1)x^4+(a+1)x^3+(a^3+2a^2+a+1)x^2+(2a^3+a^2+2a+2)x+2a^3+a+2$
- $y^2=(2a^3+1)x^6+(a^3+a+2)x^5+(2a^3+2a+1)x^4+(2a^3+a+1)x^3+(2a^3+2a+1)x^2+(a^3+a+2)x+2a^3+1$
- $y^2=(a^3+a+2)x^6+(2a^3+1)x^5+(a^3+2)x^4+(a^3+2a+2)x^3+(a^3+2)x^2+(2a^3+1)x+a^3+a+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{4}}$.
Endomorphism algebra over $\F_{3^{4}}$The isogeny class factors as 1.81.as $\times$ 1.81.ao 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.