Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 5 x + 6 x^{2} - 95 x^{3} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.0277633608662$, $\pm0.638903305800$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{17})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is simple but not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $268$ | $125424$ | $44890000$ | $16911670464$ | $6130382929828$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $15$ | $349$ | $6540$ | $129769$ | $2475825$ | $47022118$ | $893817555$ | $16983608209$ | $322686090420$ | $6131061381229$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=10 x^6+16 x^5+15 x^4+9 x^3+8 x^2+2 x+14$
- $y^2=2 x^6+12 x^5+2 x^4+15 x^3+16 x^2+11 x+2$
- $y^2=5 x^6+10 x^5+13 x^3+x^2+7 x+17$
- $y^2=14 x^6+12 x^5+15 x^4+2 x^3+17 x^2+6 x+3$
- $y^2=6 x^6+18 x^5+17 x^4+3 x^2+8 x+13$
- $y^2=2 x^6+3 x^5+5 x^4+17 x^3+3 x^2+16 x+15$
- $y^2=2 x^6+2 x^3+15$
- $y^2=12 x^5+15 x^4+13 x^3+6 x^2+15 x+9$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19^{3}}$.
Endomorphism algebra over $\F_{19}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{17})\). |
| The base change of $A$ to $\F_{19^{3}}$ is 1.6859.age 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$ |
Base change
This is a primitive isogeny class.