Invariants
| Base field: | $\F_{3^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x + 11 x^{2} - 36 x^{3} + 81 x^{4}$ |
| Frobenius angles: | $\pm0.153401645225$, $\pm0.570422193164$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.513040.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
This isogeny class is simple and geometrically 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $53$ | $7049$ | $503924$ | $42752185$ | $3534063373$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $88$ | $690$ | $6516$ | $59846$ | $533206$ | $4782854$ | $43059236$ | $387470850$ | $3486701128$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a+2)x^6+(a+2)x^5+(a+1)x^4+x^3+(2a+2)x^2+(2a+1)x+2$
- $y^2=(a+1)x^6+x^5+2ax^3+(a+2)x^2+2ax+2a+2$
- $y^2=(a+1)x^6+(2a+1)x^5+2ax^4+x^3+ax^2+ax+a$
- $y^2=(2a+1)x^6+(2a+1)x^5+ax^3+2ax^2+2x+a+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{2}}$.
Endomorphism algebra over $\F_{3^{2}}$| The endomorphism algebra of this simple isogeny class is 4.0.513040.2. |
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.e_l | $2$ | 2.81.g_af |