Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 5 x + 9 x^{2} )( 1 - x + 9 x^{2} )$ |
$1 - 6 x + 23 x^{2} - 54 x^{3} + 81 x^{4}$ | |
Frobenius angles: | $\pm0.186429498677$, $\pm0.446699620962$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $12$ |
Isomorphism classes: | 20 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $45$ | $7425$ | $559440$ | $42953625$ | $3493462725$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $92$ | $766$ | $6548$ | $59164$ | $533582$ | $4789516$ | $43044068$ | $387360334$ | $3486664652$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a+1)x^6+(a+1)x^5+2ax^4+2ax^3+(2a+2)x^2+x+2a+1$
- $y^2=(2a+1)x^6+(a+2)x^5+(a+1)x^4+2x^3+(2a+1)x^2+a+2$
- $y^2=(2a+2)x^6+(a+2)x^5+(2a+2)x^4+(a+1)x^3+(2a+2)x^2+(a+2)x+2a+2$
- $y^2=x^6+x^5+ax^4+(2a+1)x^3+ax^2+x+a+2$
- $y^2=(2a+1)x^6+(a+2)x^5+ax^4+(a+1)x^3+(a+2)x^2+(a+1)x+2a$
- $y^2=x^6+(2a+1)x^5+x^4+2x^3+x^2+(2a+1)x+1$
- $y^2=(2a+1)x^6+(2a+1)x^5+2x^4+x^3+x^2+(2a+1)x+a+2$
- $y^2=(2a+1)x^6+x^5+(2a+2)x^4+x^2+2x+a$
- $y^2=(a+2)x^6+(a+1)x^5+(2a+2)x^3+(a+1)x^2+ax+2a+1$
- $y^2=(2a+1)x^6+x^5+x^4+(2a+2)x^3+(a+1)x^2+(2a+1)x+a$
- $y^2=ax^6+x^5+(a+1)x^4+x^2+2x+2a+1$
- $y^2=(a+2)x^6+(a+2)x^5+(a+1)x^4+(2a+2)x^3+(2a+2)x^2+(a+2)x+2a+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{2}}$.
Endomorphism algebra over $\F_{3^{2}}$The isogeny class factors as 1.9.af $\times$ 1.9.ab 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 |
---|---|---|
2.9.ae_n | $2$ | 2.81.k_br |
2.9.e_n | $2$ | 2.81.k_br |
2.9.g_x | $2$ | 2.81.k_br |