Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 8 x + 19 x^{2} )( 1 - 4 x + 19 x^{2} )$ |
| $1 - 12 x + 70 x^{2} - 228 x^{3} + 361 x^{4}$ | |
| Frobenius angles: | $\pm0.130073469147$, $\pm0.348268167089$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $12$ |
| Isomorphism classes: | 60 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $192$ | $129024$ | $47791296$ | $17020846080$ | $6129255605952$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $358$ | $6968$ | $130606$ | $2475368$ | $47043286$ | $893917592$ | $16984025566$ | $322689710792$ | $6131069428678$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=12x^6+11x^5+9x^4+15x^3+9x^2+11x+12$
- $y^2=15x^6+2x^5+15x^4+16x^3+2x^2+11x+5$
- $y^2=11x^6+3x^5+x^4+18x^3+17x^2+6$
- $y^2=x^6+4x^5+10x^4+12x^3+10x^2+4x+1$
- $y^2=17x^6+15x^5+7x^4+13x^3+9x^2+8x+6$
- $y^2=17x^6+16x^5+11x^4+17x^3+17x^2+17x+8$
- $y^2=10x^6+10x^5+15x^4+6x^3+10x^2+15x+10$
- $y^2=11x^5+3x^4+5x^3+3x^2+11x$
- $y^2=15x^6+2x^5+7x^4+17x^3+7x^2+2x+15$
- $y^2=7x^6+13x^5+11x^4+9x^3+x^2+10x+7$
- $y^2=8x^6+11x^5+16x^4+11x^3+4x^2+9x+12$
- $y^2=8x^6+4x^5+5x^4+8x^3+6x^2+5x+12$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19}$.
Endomorphism algebra over $\F_{19}$| The isogeny class factors as 1.19.ai $\times$ 1.19.ae 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.