Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 4 x + 23 x^{2} - 114 x^{3} + 299 x^{4} - 676 x^{5} + 2197 x^{6}$ |
| Frobenius angles: | $\pm0.127815665289$, $\pm0.537009370572$, $\pm0.580348951049$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.435188672.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1726$ | $5719964$ | $10002510022$ | $22918500076784$ | $51510363716704366$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $200$ | $2068$ | $28092$ | $373640$ | $4833692$ | $62736446$ | $815778332$ | $10605018454$ | $137858657700$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=2 x^8+4 x^7+4 x^6+10 x^5+2 x^4+10 x^3+4 x^2+6 x+6$
- $y^2=2 x^8+8 x^7+5 x^5+9 x^4+12 x^2+3 x+11$
- $y^2=2 x^8+7 x^7+5 x^6+12 x^5+3 x^4+11 x^3+3 x^2+11 x+9$
- $y^2=2 x^8+8 x^7+2 x^6+8 x^5+9 x^4+3 x^3+11 x^2+10 x+4$
- $y^2=2 x^8+5 x^7+4 x^6+3 x^5+11 x^4+6 x^3+x^2+11$
- $y^2=x^8+x^7+x^6+4 x^5+5 x^4+x^3+6 x^2+6 x+7$
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 6.0.435188672.1. |
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.e_x_ek | $2$ | (not in LMFDB) |