Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 11 x + 55 x^{2} + 200 x^{3} + 715 x^{4} + 1859 x^{5} + 2197 x^{6}$ |
| Frobenius angles: | $\pm0.409275130495$, $\pm0.846803799211$, $\pm0.871973817905$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.424587712.1 |
| Galois group: | $S_4\times C_2$ |
| 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: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $5038$ | $4524124$ | $11187342496$ | $23291077080304$ | $50824685566304518$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $25$ | $159$ | $2314$ | $28551$ | $368665$ | $4833732$ | $62732513$ | $815868255$ | $10604127718$ | $137858178999$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobian of 1 hyperelliptic curve, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^8+6 x^6+6 x^5+9 x^4+8 x^3+2 x+10$
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 6.0.424587712.1. |
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 |
|---|---|---|
| 3.13.al_cd_ahs | $2$ | (not in LMFDB) |