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 contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+13x^5+7x^4+4x^3+9x^2+4x+15$
- $y^2=13x^6+9x^5+6x^4+2x^3+6x^2+9x+13$
- $y^2=14x^6+11x^5+2x^4+2x^2+11x+14$
- $y^2=12x^6+8x^5+4x^4+6x^3+4x^2+8x+12$
- $y^2=x^6+17x^3+11$
- $y^2=10x^6+11x^5+12x^4+5x^3+12x^2+11x+10$
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.