Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 4 x + 28 x^{2} - 52 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.343002591537$, $\pm0.474113902873$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.133376.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $142$ | $36068$ | $5088286$ | $810375824$ | $137397711982$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $210$ | $2314$ | $28374$ | $370050$ | $4826370$ | $62752210$ | $815724894$ | $10604560042$ | $137859164050$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=7 x^6+8 x^5+2 x^4+3 x^2+11 x+6$
- $y^2=7 x^6+6 x^5+7 x^4+x^3+8 x^2+5$
- $y^2=11 x^5+2 x^4+10 x^3+7 x^2+5 x+10$
- $y^2=5 x^6+9 x^4+7 x^3+10 x^2+7 x+4$
- $y^2=7 x^6+7 x^5+11 x^4+3 x^2+5 x+11$
- $y^2=6 x^6+11 x^5+2 x^4+9 x^3+8 x^2+x+3$
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 4.0.133376.1. |
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.e_bc | $2$ | 2.169.bo_bbe |