Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 7 x + 19 x^{2} )( 1 - 4 x + 19 x^{2} )$ |
| $1 - 11 x + 66 x^{2} - 209 x^{3} + 361 x^{4}$ | |
| Frobenius angles: | $\pm0.203259864187$, $\pm0.348268167089$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $6$ |
| Isomorphism classes: | 32 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $208$ | $134784$ | $48577984$ | $17093306880$ | $6133488510448$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $9$ | $373$ | $7080$ | $131161$ | $2477079$ | $47043286$ | $893874501$ | $16983666481$ | $322687757400$ | $6131061600853$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=13 x^6+x^5+5 x^4+15 x^3+18 x^2+7 x$
- $y^2=18 x^6+9 x^5+3 x^4+9 x^3+8 x^2+18 x+2$
- $y^2=4 x^5+6 x^4+10 x^3+15 x^2+3 x+3$
- $y^2=10 x^6+4 x^5+x^4+3 x^3+17 x^2+8 x+9$
- $y^2=18 x^6+8 x^5+17 x^4+12 x^3+18 x^2+15 x+15$
- $y^2=15 x^6+16 x^5+7 x^4+14 x^3+14 x^2+9$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19}$.
Endomorphism algebra over $\F_{19}$| The isogeny class factors as 1.19.ah $\times$ 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.