Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 7 x + 19 x^{2} )( 1 - 3 x + 19 x^{2} )$ |
$1 - 10 x + 59 x^{2} - 190 x^{3} + 361 x^{4}$ | |
Frobenius angles: | $\pm0.203259864187$, $\pm0.388176076177$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $8$ |
Isomorphism classes: | 14 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $221$ | $137241$ | $48439664$ | $17046567369$ | $6131185293701$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $380$ | $7060$ | $130804$ | $2476150$ | $47049446$ | $893923810$ | $16983708964$ | $322686743980$ | $6131056717100$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+8x^5+15x^4+17x^3+15x^2+8x+2$
- $y^2=9x^6+14x^5+16x^4+x^3+6x^2+4x+3$
- $y^2=3x^6+17x^5+10x^4+10x^3+10x^2+17x+3$
- $y^2=2x^6+11x^5+17x^4+8x^3+17x^2+11x+2$
- $y^2=13x^6+x^5+4x^4+8x^3+10x^2+9x+3$
- $y^2=10x^6+4x^5+15x^4+14x^3+18x^2+12x+14$
- $y^2=18x^6+12x^5+13x^4+11x^2+4x+5$
- $y^2=2x^6+17x^5+2x^4+8x^3+2x^2+17x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19}$.
Endomorphism algebra over $\F_{19}$The isogeny class factors as 1.19.ah $\times$ 1.19.ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.