Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 7 x + 31 x^{2} - 91 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.171237405357$, $\pm0.464284369344$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.589541.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $3$ |
| Isomorphism classes: | 3 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $103$ | $30797$ | $4908259$ | $812332469$ | $138024863248$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $183$ | $2233$ | $28443$ | $371742$ | $4834623$ | $62772577$ | $815717811$ | $10604298439$ | $137858354518$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):
- $y^2=x^5+5 x^4+9 x^3+10 x^2+8 x+11$
- $y^2=7 x^6+12 x^5+10 x^4+8 x^3+3 x^2+6 x+4$
- $y^2=11 x^6+8 x^5+12 x^4+10 x^3+12 x^2+10 x+2$
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 4.0.589541.1. |
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.h_bf | $2$ | 2.169.n_z |