Invariants
| Base field: | $\F_{3^{3}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 8 x + 27 x^{2} )^{2}$ |
| $1 - 16 x + 118 x^{2} - 432 x^{3} + 729 x^{4}$ | |
| Frobenius angles: | $\pm0.220355751984$, $\pm0.220355751984$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $10$ |
This isogeny class is not simple, not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $400$ | $518400$ | $392832400$ | $283875840000$ | $206097607210000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $710$ | $19956$ | $534158$ | $14363292$ | $387462230$ | $10460298756$ | $282427973918$ | $7625586454572$ | $205891086040550$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+2x^4+x^3+2x^2+1$
- $y^2=(2a+1)x^6+ax^5+(2a^2+2a)x^4+(2a+2)x^3+(2a^2+2a)x^2+ax+2a+1$
- $y^2=(2a^2+2a+2)x^6+(2a+1)x^5+(2a^2+2a+1)x^4+2ax^3+(2a^2+2a+1)x^2+(2a+1)x+2a^2+2a+2$
- $y^2=(2a^2+1)x^6+2a^2x^4+2a^2x^2+2a^2+1$
- $y^2=(2a^2+2)x^6+(2a^2+2a+2)x^4+(2a^2+2a+2)x^2+2a^2+2$
- $y^2=ax^6+(a^2+a+2)x^4+(a^2+2)x^3+(a^2+a+2)x^2+a$
- $y^2=2x^6+(2a^2+2a+2)x^5+x^4+x^2+(2a^2+2a+2)x+2$
- $y^2=(2a^2+2a)x^6+(2a^2+2a+2)x^4+(2a^2+2a+2)x^2+2a^2+2a$
- $y^2=ax^6+2x^4+(2a+2)x^3+2x^2+a$
- $y^2=ax^6+(2a^2+a+2)x^4+(2a^2+2)x^3+(2a^2+a+2)x^2+a$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{3}}$.
Endomorphism algebra over $\F_{3^{3}}$| The isogeny class factors as 1.27.ai 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$ |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{3}}$.
| Subfield | Primitive Model |
| $\F_{3}$ | 2.3.ab_ac |
| $\F_{3}$ | 2.3.c_h |