Invariants
Base field: | $\F_{19}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 7 x + 19 x^{2} )( 1 - 5 x + 19 x^{2} )$ |
$1 - 12 x + 73 x^{2} - 228 x^{3} + 361 x^{4}$ | |
Frobenius angles: | $\pm0.203259864187$, $\pm0.305569972467$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $8$ |
Isomorphism classes: | 12 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $195$ | $131625$ | $48550320$ | $17134547625$ | $6138171800475$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $8$ | $364$ | $7076$ | $131476$ | $2478968$ | $47044582$ | $893831912$ | $16983417316$ | $322687524764$ | $6131066208124$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+7x^5+7x+3$
- $y^2=8x^6+9x^5+13x^4+4x^3+12x^2+17x+12$
- $y^2=16x^6+x^5+10x^4+17x^3+18x^2+4x+5$
- $y^2=13x^6+9x^5+13x^4+7x^3+12x^2+17x+10$
- $y^2=14x^6+x^5+2x^4+15x^3+13x^2+9x+2$
- $y^2=3x^6+17x^5+x^4+10x^3+x^2+17x+3$
- $y^2=10x^6+8x^5+8x^4+18x^3+12x^2+18x+10$
- $y^2=13x^6+15x^5+7x^4+2x^3+7x^2+15x+13$
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.af 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.