Calculator Input
Use one stacked page layout. The calculator grid below expands to three columns on large screens, two on smaller screens, and one on mobile.
Example Data Table
Use this sample configuration to verify the workflow. The listed relation rows are [1,-1,0,0], [0,1,-1,0], and [2,0,0,-2].
| Prime Divisor | Degree | Coefficient in D | Coefficient in E |
|---|---|---|---|
| P1 | 1 | 2 | 0 |
| P2 | 1 | -1 | 1 |
| P3 | 1 | 3 | 1 |
| P4 | 1 | 0 | 0 |
Formula Used
This tool works with a finitely generated free divisor group Div = Z^n, where each basis element is a chosen prime divisor. The entered principal relation rows generate Prin. The divisor class group is then computed as Cl = Div / Prin.
Let A be the integer matrix whose columns generate the principal subgroup. The calculator computes a Smith normal form
U A V = diag(d1, d2, ..., dr, 0, ..., 0)
with each nonzero diagonal entry dividing the next. That gives
Cl ≅ Z^(n-r) ⊕ Z/d1Z ⊕ ... ⊕ Z/drZ
after omitting any trivial factor Z/1Z. For a divisor vector D, the transformed coordinates U(D) determine the class signature. The degree is computed by
deg(D) = Σ ai · deg(Pi)
where D = Σ ai Pi. Two divisors are linearly equivalent exactly when D − E maps to the zero class.
How to Use This Calculator
- Enter your prime divisor labels in the basis order you want.
- Provide the matching degree vector. Use all ones when appropriate.
- Type the coefficients of divisor D in the same order.
- Optionally enter divisor E to test linear equivalence.
- Add one principal divisor relation per line.
- Press the compute button to build the quotient group.
- Read the SNF diagonal, torsion signature, and free coordinates.
- Use CSV or PDF export for documentation or coursework.
Frequently Asked Questions
1. What does this calculator actually compute?
It computes the divisor class group of the finitely presented divisor model you enter. Your basis divisors define Div, and the relation rows generate the subgroup of principal divisors used for the quotient.
2. Does it solve divisor class groups for every scheme automatically?
No. General class group computation depends on the variety, ring, or curve structure. This page is a rigorous presentation-based calculator, not a universal engine for arbitrary algebraic varieties.
3. Why is Smith normal form used here?
Smith normal form diagonalizes the integer relation matrix using unimodular changes of basis. That immediately reveals the free rank, torsion factors, and canonical residue data of the quotient group.
4. What does the class order mean?
The class order is the smallest positive integer n such that n[D] becomes principal. If any free coordinate is nonzero, the order is infinite.
5. Why can a divisor have infinite order?
If the quotient has a free part, some classes survive in a copy of Z. Any nonzero component in that free part forces infinite order for the divisor class.
6. What is the difference between principal and equivalent?
A divisor is principal when its own class is zero. Two divisors are equivalent when their difference is principal, so they represent the same element of the quotient group.
7. How should I choose the degree vector?
Use the degree attached to each basis prime divisor in your model. For many instructional examples on curves, every chosen prime divisor may have degree one.
8. What if I leave the relation box empty?
Then the principal subgroup is trivial. The divisor class group equals the full free divisor group on your chosen basis, so every nonzero divisor class has infinite order.