Invariants
| Base field: | $\F_{3^{3}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 10 x + 27 x^{2} )( 1 - 4 x + 27 x^{2} )$ |
| $1 - 14 x + 94 x^{2} - 378 x^{3} + 729 x^{4}$ | |
| Frobenius angles: | $\pm0.0877398280459$, $\pm0.374235869875$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $18$ |
| Isomorphism classes: | 120 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $432$ | $525312$ | $388788336$ | $282088341504$ | $205770439178352$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $14$ | $722$ | $19754$ | $530798$ | $14340494$ | $387395522$ | $10460500106$ | $282431229086$ | $7625604068558$ | $205891138890482$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a^2+2a)x^5+(2a^2+2a+2)x^4+(2a^2+1)x^3+(a+2)x^2+(2a+1)x+a^2+a+2$
- $y^2=(2a^2+a)x^6+(a^2+2a+2)x^5+ax^4+(2a^2+2a)x^3+ax^2+(a^2+2a+2)x+2a^2+a$
- $y^2=(2a^2+a)x^6+(2a^2+a+2)x^5+2a^2x^4+(a^2+1)x^3+(a^2+2a+2)x^2+(a^2+2a+1)x+2a^2+1$
- $y^2=a^2x^6+(a+2)x^5+(a^2+a+1)x^4+(2a^2+1)x^3+(a+1)x^2+(2a^2+1)x+2a^2+2a$
- $y^2=2ax^6+(2a^2+a+1)x^5+ax^4+2x^3+ax^2+(2a^2+a+1)x+2a$
- $y^2=(2a^2+a)x^6+(a^2+a+1)x^5+ax^4+(2a^2+1)x^3+(a+2)x^2+(a^2+a)x+2a^2+2a$
- $y^2=x^6+(a^2+a+2)x^5+(a+2)x^4+(2a^2+2)x^3+(2a^2+a+2)x^2+(2a+2)x$
- $y^2=a^2x^5+2x^4+x^3+(a+2)x^2+2a^2x+a^2+2a+2$
- $y^2=2a^2x^6+(2a^2+a+1)x^5+ax^4+(a+1)x^3+(2a^2+2a+2)x^2+(a^2+2a)x+2a^2+a$
- $y^2=(2a^2+a)x^6+(a^2+a+1)x^5+(a^2+2a)x^4+(a^2+a)x^3+(2a^2+a+1)x^2+(2a+2)x+a$
- $y^2=(2a^2+2a+2)x^6+(2a^2+a+2)x^5+(a^2+2a)x^4+(a^2+2)x^3+(2a+2)x^2+2a^2+2a+2$
- $y^2=ax^6+(a^2+a+2)x^5+(a^2+2)x^4+(a^2+2a)x^2+2ax+2a^2+a$
- $y^2=(2a^2+a)x^6+(2a^2+1)x^5+x^4+(2a^2+a+1)x^3+x^2+(2a^2+1)x+2a^2+a$
- $y^2=ax^6+(a+2)x^5+(a+1)x^4+(2a^2+1)x^3+(a^2+a)x^2+(a^2+1)x+a^2+2a+2$
- $y^2=ax^6+(2a^2+a+1)x^5+(2a^2+a+1)x^4+(a^2+a+2)x^3+(2a^2+2)x^2+(2a^2+1)x+2$
- $y^2=(2a^2+2a)x^6+(2a+1)x^5+ax^4+(a^2+1)x^3+(a^2+2a+2)x^2+a^2x+2a^2$
- $y^2=(2a^2+2a)x^6+a^2x^5+(a^2+a)x^4+(a^2+2)x^3+(2a^2+2a+1)x^2+(2a+1)x+a+1$
- $y^2=(a+1)x^6+(2a^2+2a+1)x^5+(2a^2+2a+1)x^4+(a^2+1)x^3+(2a^2+a+1)x^2+(2a^2+a)x+2$
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.ak $\times$ 1.27.ae 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.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.27.ag_o | $2$ | 2.729.ai_ale |
| 2.27.g_o | $2$ | 2.729.ai_ale |
| 2.27.o_dq | $2$ | 2.729.ai_ale |