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

- 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 : setRandomSeed 0
     -- setting random seed to 0

    o1 = 0

    i2 : kk=ZZ/101;

    i3 : g=1;

    i4 : rNP=randNicePencil(kk,g);

    i5 : f=(cliffordModule(rNP.matFact1,rNP.matFact2,rNP.baseRing)).hyperellipticBranchEquation

              3       2 2        3      4
    o5 = - 12s t - 50s t  - 16s*t  + 47t

    o5 : kk[s, t]

    i6 : L=randomLineBundle(0,f);

    i7 : orderInPic L

    o7 = 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.

## See also

- randomLineBundle -- a random balanced line bundle on the hyperelliptic curve
- vectorBundleOnE -- creates a VectorBundleOnE, represented as a matrix factorization
- VectorBundleOnE -- vector bundle on a hyperelliptic curve E
- yAction -- defines a vector bundle on E

## Ways to use orderInPic:

- `orderInPic(Matrix)`
- `orderInPic(VectorBundleOnE)`

## For the programmer

The object orderInPic is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/PencilsOfQuadrics.m2:3032:0.