Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - x + 19 x^{2} )( 1 + x + 19 x^{2} )$ |
| $1 + 37 x^{2} + 361 x^{4}$ | |
| Frobenius angles: | $\pm0.463406802480$, $\pm0.536593197520$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $14$ |
| Isomorphism classes: | 26 |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $399$ | $159201$ | $47056464$ | $16815605625$ | $6131068282479$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $436$ | $6860$ | $129028$ | $2476100$ | $47067046$ | $893871740$ | $16983247108$ | $322687697780$ | $6131070307156$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):
- $y^2=9 x^6+5 x^5+18 x^4+16 x^3+9 x^2+16 x+2$
- $y^2=18 x^6+10 x^5+17 x^4+13 x^3+18 x^2+13 x+4$
- $y^2=10 x^6+7 x^5+6 x^4+15 x^3+9 x^2+11 x+10$
- $y^2=x^6+14 x^5+12 x^4+11 x^3+18 x^2+3 x+1$
- $y^2=17 x^6+11 x^5+16 x^4+10 x^3+5 x^2+x+17$
- $y^2=15 x^6+3 x^5+13 x^4+x^3+10 x^2+2 x+15$
- $y^2=12 x^6+17 x^5+10 x^4+9 x^3+10 x^2+17 x+12$
- $y^2=5 x^6+15 x^5+x^4+18 x^3+x^2+15 x+5$
- $y^2=18 x^6+x^5+18 x^4+14 x^3+14 x^2+6 x+8$
- $y^2=17 x^6+2 x^5+17 x^4+9 x^3+9 x^2+12 x+16$
- $y^2=11 x^6+13 x^5+14 x^4+16 x^3+2 x^2+15 x+11$
- $y^2=3 x^6+7 x^5+9 x^4+13 x^3+4 x^2+11 x+3$
- $y^2=8 x^6+3 x^5+12 x^4+6 x^3+8 x^2+14 x+8$
- $y^2=16 x^6+6 x^5+5 x^4+12 x^3+16 x^2+9 x+16$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19^{2}}$.
Endomorphism algebra over $\F_{19}$| The isogeny class factors as 1.19.ab $\times$ 1.19.b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
| The base change of $A$ to $\F_{19^{2}}$ is 1.361.bl 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
Base change
This is a primitive isogeny class.