Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 5 x + 45 x^{2} - 131 x^{3} + 585 x^{4} - 845 x^{5} + 2197 x^{6}$ |
| Frobenius angles: | $\pm0.351325929916$, $\pm0.430818226707$, $\pm0.491256107388$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.256443607.1 |
| Galois group: | $A_4\times C_2$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1847$ | $7035223$ | $11391610763$ | $22929881218231$ | $50893829106093947$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $9$ | $235$ | $2355$ | $28107$ | $369169$ | $4828159$ | $62765348$ | $815731827$ | $10604437989$ | $137858754775$ |
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:
- $y^2=2 x^7+x^5+12 x^4+12 x^3+2 x^2+4 x+4$
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 6.0.256443607.1. |
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.f_bt_fb | $2$ | (not in LMFDB) |