Invariants
| Base field: | $\F_{3^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 5 x + 9 x^{2} )( 1 - 2 x + 9 x^{2} )$ |
| $1 - 7 x + 28 x^{2} - 63 x^{3} + 81 x^{4}$ | |
| Frobenius angles: | $\pm0.186429498677$, $\pm0.391826552031$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $3$ |
| Isomorphism classes: | 15 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $40$ | $7200$ | $574240$ | $43574400$ | $3486260200$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $3$ | $89$ | $786$ | $6641$ | $59043$ | $532142$ | $4788507$ | $43059041$ | $387396354$ | $3486562649$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2ax^6+2x^5+(2a+1)x^3+(a+1)x^2+(a+1)x+2a$
- $y^2=ax^6+(a+1)x^5+ax^4+2x^3+2x^2+(2a+2)x$
- $y^2=x^5+(2a+2)x^4+ax^3+2ax^2+ax$
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.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.ad_i | $2$ | 2.81.h_cm |
| 2.9.d_i | $2$ | 2.81.h_cm |
| 2.9.h_bc | $2$ | 2.81.h_cm |