Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 3 x + 14 x^{2} + 39 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.397836407734$, $\pm0.761179122709$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.3677868.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $226$ | $32092$ | $4866232$ | $822582144$ | $137037292426$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $17$ | $189$ | $2216$ | $28801$ | $369077$ | $4826202$ | $62769017$ | $815723809$ | $10604631512$ | $137857495509$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=2 x^5+9 x^4+7 x^3+8 x^2+4 x+6$
- $y^2=9 x^6+12 x^5+10 x^4+8 x^3+6 x^2+6 x+6$
- $y^2=2 x^6+5 x^5+3 x^4+9 x^3+5 x^2+9 x+10$
- $y^2=x^6+12 x^5+11 x^4+5 x^3+3 x^2+6$
- $y^2=8 x^6+10 x^5+9 x^4+11 x^2+8 x+10$
- $y^2=6 x^6+3 x^4+6 x^2+4 x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13}$.
Endomorphism algebra over $\F_{13}$| The endomorphism algebra of this simple isogeny class is 4.0.3677868.1. |
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 |
|---|---|---|
| 2.13.ad_o | $2$ | 2.169.t_lo |