Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 10 x + 77 x^{2} - 354 x^{3} + 1463 x^{4} - 3610 x^{5} + 6859 x^{6}$ |
| Frobenius angles: | $\pm0.278329722546$, $\pm0.292576990832$, $\pm0.532162528353$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.3541000640.1 |
| Galois group: | $S_4\times C_2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4426$ | $54767324$ | $334549226038$ | $2224075834670960$ | $15199177069152382426$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $416$ | $7108$ | $130956$ | $2479040$ | $47040308$ | $893714762$ | $16983060572$ | $322689073678$ | $6131077556316$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 3 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=18 x^8+14 x^7+x^6+16 x^5+4 x^4+3 x^3+4 x^2+17 x+13$
- $y^2=18 x^8+17 x^7+10 x^6+9 x^5+12 x^4+15 x^3+x^2+12 x+3$
- $y^2=18 x^8+14 x^7+2 x^6+16 x^5+16 x^4+2 x^3+4 x^2+12 x+14$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19}$.
Endomorphism algebra over $\F_{19}$| The endomorphism algebra of this simple isogeny class is 6.0.3541000640.1. |
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 |
|---|---|---|
| 3.19.k_cz_nq | $2$ | (not in LMFDB) |