Invariants
| Base field: | $\F_{3^{4}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 17 x + 81 x^{2} )^{2}$ |
| $1 - 34 x + 451 x^{2} - 2754 x^{3} + 6561 x^{4}$ | |
| Frobenius angles: | $\pm0.106600758076$, $\pm0.106600758076$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4225$ | $41409225$ | $281600035600$ | $1852761401001225$ | $12157750704111390625$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $48$ | $6308$ | $529878$ | $43040708$ | $3486808848$ | $282430439198$ | $22876805821008$ | $1853020342954628$ | $150094636834096758$ | $12157665472705262948$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2ax^6+ax^5+2ax^4+ax^2+ax+a$
- $y^2=ax^6+(a^3+2a+2)x^4+(a^3+a+2)x^3+(a^3+2a+2)x^2+a$
- $y^2=(a^3+2a+2)x^6+(2a^3+a+1)x^5+2ax^3+(a^3+2a+2)x+2a$
- $y^2=ax^6+(2a^3+a+1)x^4+(2a^3+1)x^3+(2a^3+a+1)x^2+a$
- $y^2=(2a^3+a+1)x^6+(a^3+2a+2)x^5+2ax^3+(2a^3+a+1)x+2a$
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.ar 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-35}) \)$)$ |
Base change
This is a primitive isogeny class.