Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 2 x + 19 x^{2} )( 1 + x + 19 x^{2} )$ |
| $1 - x + 36 x^{2} - 19 x^{3} + 361 x^{4}$ | |
| Frobenius angles: | $\pm0.426318466621$, $\pm0.536593197520$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $7$ |
| Isomorphism classes: | 41 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $378$ | $158004$ | $47396664$ | $16843226400$ | $6128166765798$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $19$ | $433$ | $6910$ | $129241$ | $2474929$ | $47058946$ | $893888371$ | $16983477361$ | $322687684330$ | $6131064928753$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which all are hyperelliptic):
- $y^2=12 x^6+2 x^5+4 x^4+9 x^3+17 x^2+4 x+2$
- $y^2=5 x^6+12 x^5+17 x^4+16 x^3+4 x^2+12 x+17$
- $y^2=3 x^6+17 x^5+6 x^4+17 x^3+7 x^2+18 x+3$
- $y^2=4 x^6+2 x^5+9 x^4+10 x^3+14 x^2+17 x+5$
- $y^2=12 x^6+2 x^5+x^4+15 x^3+14 x^2+13 x+6$
- $y^2=9 x^6+5 x^5+6 x^4+14 x^3+16 x+9$
- $y^2=17 x^6+16 x^5+5 x^4+3 x^3+16 x^2+18 x+9$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19}$.
Endomorphism algebra over $\F_{19}$| The isogeny class factors as 1.19.ac $\times$ 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: |
Base change
This is a primitive isogeny class.