2222.sagews

Algebraic Number Theory

MAA PREP Workshop, Summer 2010

Extending Problem 3 of the Computational Excercises: Possible Explorations

Introduction to Sage, Pari, Class Numbers, Discriminants, and Continued Fractions

The goal of this worksheet is to orient students to Sage and Pari - in particular, how to created pari outputs you can manipulate in sage.  In addition, it gives them some experience looking at class numbers, regulators, and discriminants.  It guides them through making a few initial conjectures, examining them with Pari, and looking up related literature.

0. Consider the quadratic number field Q(sqrt(d)).  Review the definition of class number, regulator and discriminant.

The following code uses pari commands to generate vectors of discriminant, class number, and regulator values in Q(sqrt(d)) for the first 100 negative values of d congruent to 1 mod 4.

• LN1 is the list of class numbers
• RN1 is the list of regulators
• XN1 is the list of discriminants

pari("d=1");
pari("l=[]");
pari("x=[]");
pari("r=[]");
for i in range(1,100):
    pari("l=concat(l,quadclassunit(-4*d+1)[1])")
    pari("r=concat(r,quadclassunit(-4*d+1)[4])")
    pari("x=concat(x,-4*d+1)")
    pari("d=d++")
LN1=pari("l");
XN1=pari("x");
RN1=pari("r");

Class Number vs. Discriminant

d = 1 mod 4

point([(XN1[i],LN1[i]) for i in range(0,99)],color='red')

Regulator vs. Discriminant

d = 1 mod 4

point([(XN1[i],RN1[i]) for i in range(0,99)],color='red')

1. Write code to calculate:

• The class number, regulator, and discriminant for the first 100 negative values of d congruent to 2 or 3 mod 4
• The class number, regulator, and discriminant for the first 100 positive values of d congruent to 2 or 3 mod 4
• The class number, regulator, and discriminant for the first 100 positive values of d congruent to 1 mod 4

2. Graph the values of the regulator vs. discriminant and class number vs. discriminant for positive and for negative values of d.

 

2. (continued)

• What patterns do you notice?
• Is there any difference in the pattern for positive vs. negative values of d?
• Why is the regulator always 1 for negative values of d?
• Does there appear to be a finite or infinite number of negative values of d for which the class number is 1?  How about positive values of d?  
• For an arbitrary positive integer k, does there appear to be a finite or infinite number of negative values of d for which the class number is k?  How about positive values of d?  Make a conjecture and then look up the answer in the literature.

3. Does there appear to be an interesting relationship between the discriminant and the smallest possible value of the regulator? 

• Look up Pell's equation and the method of using the continued fraction representation of sqrt(d) to determine the fundamental unit.
• Determine a way to use this information to calculate a lower bound for the regulator for various values of d.
• Graph your lower bound for the regulator vs. the discriminant.  Do you see any interesting patterns?   

4.  Examine the relationship between the number of prime factors of the discriminant and the power of 2 dividing the class number.

• Graph the number of prime factors of the discriminant vs. the highest power of 2 dividing the class number for various values of d
  
• What patterns do you observe?
• Do a literature search or subsequent experimentation to understand these patterns more.
  

5. Suggest your own question regarding class number, regulator, or discriminant to investigate.