Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 4 x + 19 x^{2} )^{2}$ |
| $1 - 8 x + 54 x^{2} - 152 x^{3} + 361 x^{4}$ | |
| Frobenius angles: | $\pm0.348268167089$, $\pm0.348268167089$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $14$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $256$ | $147456$ | $49336576$ | $17045913600$ | $6120359332096$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $406$ | $7188$ | $130798$ | $2471772$ | $47019526$ | $893848548$ | $16983971038$ | $322689770412$ | $6131066796406$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):
- $y^2=18 x^6+2 x^4+2 x^2+18$
- $y^2=3 x^6+12 x^4+12 x^2+3$
- $y^2=8 x^6+15 x^5+6 x^4+x^3+6 x^2+15 x+8$
- $y^2=16 x^6+16 x^4+x^3+16 x^2+16$
- $y^2=x^6+7 x^5+2 x^4+13 x^3+2 x^2+7 x+1$
- $y^2=2 x^6+13 x^3+3$
- $y^2=8 x^6+4 x^5+8 x^4+12 x^3+8 x^2+4 x+8$
- $y^2=3 x^6+13 x^5+18 x^4+7 x^3+18 x^2+13 x+3$
- $y^2=8 x^6+4 x^5+3 x^4+2 x^3+15 x^2+5 x+12$
- $y^2=10 x^5+16 x^4+x^3+16 x^2+10 x$
- $y^2=8 x^6+14 x^4+14 x^2+8$
- $y^2=2 x^6+2 x^3+2$
- $y^2=11 x^6+10 x^4+10 x^2+11$
- $y^2=15 x^6+7 x^5+15 x^4+15 x^3+12 x^2+6 x+13$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19}$.
Endomorphism algebra over $\F_{19}$| The isogeny class factors as 1.19.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$ |
Base change
This is a primitive isogeny class.