Invariants
| Base field: | $\F_{19^{2}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 - 26 x + 361 x^{2}$ |
| Frobenius angles: | $\pm0.260146938293$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $16$ |
| Isomorphism classes: | 16 |
This isogeny class is simple and geometrically simple, not primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $1$ |
| Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $336$ | $130368$ | $47056464$ | $16983821568$ | $6131069159376$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $336$ | $130368$ | $47056464$ | $16983821568$ | $6131069159376$ | $2213314901179200$ | $799006684270354896$ | $288441413534752601088$ | $104127350297602681851984$ | $37589973457549801194193728$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which 0 are hyperelliptic):
- $y^2=x^3+a^{182} x+a^{182}$
- $y^2=x^3+a^{333} x+a^{334}$
- $y^2=x^3+a^{171} x+a^{172}$
- $y^2=x^3+a^{47} x+a^{48}$
- $y^2=x^3+a^{295} x+a^{295}$
- $y^2=x^3+5 x+5$
- $y^2=x^3+a^{75} x+a^{75}$
- $y^2=x^3+a^{228} x+a^{228}$
- $y^2=x^3+a^{98} x+a^{99}$
- $y^2=x^3+a^{345} x+a^{345}$
- $y^2=x^3+1$
- $y^2=x^3+a^{205} x+a^{205}$
- $y^2=x^3+a^{137} x+a^{138}$
- $y^2=x^3+a^{218} x+a^{218}$
- $y^2=x^3+a^{12} x+a^{12}$
- $y^2=x^3+a^{26} x+a^{27}$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19^{2}}$.
Endomorphism algebra over $\F_{19^{2}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \). |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{19^{2}}$.
| Subfield | Primitive Model |
| $\F_{19}$ | 1.19.ai |
| $\F_{19}$ | 1.19.i |