orderInPic -- order of a line bundle of degree 0 in Pic(E)

Synopsis

Usage:

N = orderInPic(L)

N = orderInPic(phi)
Inputs:
- L, an instance of the type VectorBundleOnE, a line bundle of degree 0 on E
- phi, a matrix, a matrix which is the yAction of VectorBundleOnE
Outputs:
- N, an integer, the order of L in Pic(E)

Description

Computes the order of a degree 0 line bundle L on the hyperelliptic curve E by the most naive method.

i1 : kk=ZZ/101;

i2 : g=1;

i3 : rNP=randNicePencil(kk,g);

i4 : f=(cliffordModule(rNP.matFact1,rNP.matFact2,rNP.baseRing)).hyperellipticBranchEquation

          3       2 2        3      4
o4 = - 12s t - 50s t  - 16s*t  + 47t

o4 : kk[s, t]

i5 : L=randomLineBundle(0,f);

i6 : orderInPic L

o6 = 37

Caveat

The ground field kk has to be finite. It computes the order by checking inductively whether L^k is trivial, so it may fail when kk is not finite, or has too many elements. orderInPic(phi) may not terminate if it is not the yAction of a line bundle of degree 0 on E.

Ways to use orderInPic :

orderInPic(Matrix)
orderInPic(VectorBundleOnE)

For the programmer

The object orderInPic is a method function.