Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 2 x + 19 x^{2} )^{2}$ |
| $1 - 4 x + 42 x^{2} - 76 x^{3} + 361 x^{4}$ | |
| Frobenius angles: | $\pm0.426318466621$, $\pm0.426318466621$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $11$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $324$ | $156816$ | $48525156$ | $16870892544$ | $6116807275524$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $16$ | $430$ | $7072$ | $129454$ | $2470336$ | $47050846$ | $893991184$ | $16983707614$ | $322685717488$ | $6131059550350$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 11 curves (of which all are hyperelliptic):
- $y^2=13 x^6+18 x^4+8 x^3+13 x^2+14 x+13$
- $y^2=3 x^6+10 x^5+13 x^4+14 x^3+15 x^2+15 x+3$
- $y^2=x^6+8 x^5+14 x^4+14 x^2+11 x+1$
- $y^2=9 x^6+16 x^5+7 x^4+8 x^3+7 x^2+16 x+9$
- $y^2=x^6+6 x^3+7$
- $y^2=x^5+17 x^4+12 x^3+17 x^2+x$
- $y^2=14 x^6+14 x^4+14 x^2+14$
- $y^2=x^6+9 x^3+7$
- $y^2=9 x^6+5 x^5+18 x^4+7 x^3+x^2+7 x+18$
- $y^2=7 x^6+3 x^5+3 x^4+5 x^3+7 x^2+12 x+13$
- $y^2=7 x^6+x^5+7 x^4+13 x^3+7 x^2+x+7$
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 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$ |
Base change
This is a primitive isogeny class.