Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 4 x + 9 x^{2} )( 1 - 2 x + 9 x^{2} )$ |
$1 - 6 x + 26 x^{2} - 54 x^{3} + 81 x^{4}$ | |
Frobenius angles: | $\pm0.267720472801$, $\pm0.391826552031$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $8$ |
Isomorphism classes: | 24 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $48$ | $8064$ | $600624$ | $43868160$ | $3472262448$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $98$ | $820$ | $6686$ | $58804$ | $530306$ | $4781956$ | $43046846$ | $387406180$ | $3486732578$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a+2)x^6+(2a+2)x^5+2ax^4+2x^3+2ax^2+(2a+2)x+a+2$
- $y^2=(2a+1)x^6+ax^5+(a+1)x^4+2ax^3+(a+1)x^2+ax+2a+1$
- $y^2=(a+2)x^5+(2a+2)x^4+(2a+2)x^2+(a+2)x$
- $y^2=2ax^6+(a+2)x^5+(2a+2)x^4+2ax^3+(a+1)x^2+(a+2)x+a$
- $y^2=2ax^6+2ax^5+ax^4+2x^3+(2a+1)x^2+ax+2a+1$
- $y^2=(2a+1)x^6+2x^5+2ax^4+(a+1)x^3+2ax^2+2x+2a+1$
- $y^2=2ax^6+(2a+1)x^5+x^4+2ax^3+2x^2+(2a+1)x+a$
- $y^2=2ax^6+2ax^5+ax^4+(a+1)x^3+(a+2)x^2+ax+a+2$
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.ae $\times$ 1.9.ac 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.ac_k | $2$ | 2.81.q_hi |
2.9.c_k | $2$ | 2.81.q_hi |
2.9.g_ba | $2$ | 2.81.q_hi |