Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 5 x + 19 x^{2} )^{2}$ |
| $1 - 10 x + 63 x^{2} - 190 x^{3} + 361 x^{4}$ | |
| Frobenius angles: | $\pm0.305569972467$, $\pm0.305569972467$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $6$ |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $225$ | $140625$ | $49280400$ | $17128265625$ | $6129709430625$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $388$ | $7180$ | $131428$ | $2475550$ | $47022118$ | $893763370$ | $16983472708$ | $322689305140$ | $6131076010948$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=6 x^6+13 x^5+7 x^4+4 x^3+9 x^2+4 x+15$
- $y^2=13 x^6+9 x^5+6 x^4+2 x^3+6 x^2+9 x+13$
- $y^2=14 x^6+11 x^5+2 x^4+2 x^2+11 x+14$
- $y^2=12 x^6+8 x^5+4 x^4+6 x^3+4 x^2+8 x+12$
- $y^2=x^6+17 x^3+11$
- $y^2=10 x^6+11 x^5+12 x^4+5 x^3+12 x^2+11 x+10$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19}$.
Endomorphism algebra over $\F_{19}$| The isogeny class factors as 1.19.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$ |
Base change
This is a primitive isogeny class.