2205.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

 

Problem 3: Examining the relationship between the class number, discriminant, and regulator.

The following code uses pari commands to generate a vectors of discriminant, class number, and regulator values for Q(sqrt{d}) for various values of d. 

• LN is the list of class numbers for the first 100 negative values of d congruent to 2 or 3 mod 4
• LN1 is the list of class numbers for the first 100 negative values of d congruent to 1 mod 4
• LP is the list of class numbers for the first MP positive, square-free values of d congruent to 2 or 3 mod 4
• LP1 is the list of class numbers for the first MP1 negative values of d congruent to 1 mod 4
• RN is the list of regulators for the first 100 negative values of d congruent to 2 or 3 mod 4
• RN1 is the list of regulators for the first 100 negative values of d congruent to 1 mod 4
• RP is the list of regulators for the first MP positive, square-free values of d congruent to 2 or 3 mod 4
• RP1 is the list of regulators for the first MP1 negative values of d congruent to 1 mod 4
• XN is the list of discriminants for the first 100 negative values of d congruent to 2 or 3 mod
• XN1 is the list of discriminants for the first 100 negative values of d congruent to 1 mod 4
• XP is the list of discriminants for the first MP positive, square-free values of d congruent to 2 or 3 mod 4
• XP1 is the list of discriminants for the first MP1 negative values of d congruent to 1 mod 4

pari("c=0");  
pari("l=[]");
pari("x=[]");
pari("m=0");
pari("r=[]");
for i in range(0,50):
    pari("d=4*c+2");
    pari("D=4*d");
    pari("if(issquarefree(d),l=concat(l,quadclassunit(D)[1]);x=concat(x,D);r=concat(r,quadclassunit(D)[4]);m=m++))");
    pari("d=4*c+3");
    pari("D=4*d");
    pari("if(issquarefree(d),l=concat(l,quadclassunit(D)[1]);x=concat(x,D);r=concat(r,quadclassunit(D)[4]);m=m++))");
    pari("c=c++");
LP=pari("l");
XP=pari("x");
MP=pari("m");
RP=pari("r");

pari("d=1");  #don't want to start at 0, calls 1 squarefree, but not viable D.
pari("l=[]");
pari("x=[]");
pari("m=0");
pari("r=[]");
for i in range(1,100):
    pari("D=4*d+1");
    pari("if(issquarefree(D),l=concat(l,quadclassunit(D)[1]);x=concat(x,D);r=concat(r,quadclassunit(D)[4]);m=m++)");
    pari("d=d++");
LP1=pari("l");
XP1=pari("x");
MP1=pari("m");
RP1=pari("r");

pari("d=0");
pari("l=[]");
pari("x=[]");
pari("r=[]");
for i in range(0,50):
    pari("l=concat(l,quadclassunit(-16*d-4)[1])")
    pari("r=concat(r,quadclassunit(-16*d-4)[4])")
    pari("x=concat(x,-16*d-4)")
    pari("l=concat(l,quadclassunit(-16*d-8)[1])")
    pari("r=concat(r,quadclassunit(-16*d-8)[4])")
    pari("x=concat(x,-16*d-8)")
    pari("d=d++")
LN=pari("l");
XN=pari("x");
RN=pari("r");

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 (plotted in red); d = 2,3 mod 4 (plotted in blue)

point([(XP1[i],LP1[i]) for i in range(0,(MP1))],color='red')+point([(XP[i],LP[i]) for i in range(0,(MP))])+point([(XN1[i],LN1[i]) for i in range(0,99)],color='red')+point([(XN[i],LN[i]) for i in range(0,99)])

Regulator vs. Discriminant

d = 1 mod 4 (plotted in red); d = 2,3 mod 4 (plotted in blue)

point([(XP1[i],RP1[i]) for i in range(0,(MP1))],color='red')+point([(XP[i],RP[i]) for i in range(0,(MP))])+point([(XN1[i],RN1[i]) for i in range(0,99)],color='red')+point([(XN[i],RN[i]) for i in range(0,99)])

Observations:

Appears to be an interesting logarithmic(?) bound to the minimum possible regulator value as a function of d for d>0.  Examined further below.

Using Continued Fractions to Determine a Lower Bound on the Regulator R for Real Quadratic Fields

pari("d=[]");
pari("a=[]");
pari("m=2");
pari("cnt=0");
for i in range(0,800):
    pari("if(!issquare(m),L=contfrac(sqrt(m));d=concat(d,m);cnt=cnt++;a=concat(a,(L[1]*L[2]+1+L[2]*sqrt(m))))");
    pari("m=m++");
ab=pari("a");
dp=pari("d");
CNT=pari("cnt");
pari("cnt")

773

• ab is the value a+b(sqrt(d)) where a/b is the approximation of sqrt(d) from the first two terms of the continued fraction representation of sqrt(d)
  

point([(dp[i],ab[i]) for i in range(0,(CNT-1))])

a+bsqrt(d) vs. d

pari("d=[]");
pari("a=[]");
pari("m=2");
pari("cnt=0");
for i in range(0,800):
    pari("if(issquarefree(m),L=contfrac(sqrt(m));d=concat(d,m);cnt=cnt++;a=concat(a,(L[1]*L[2]+1+L[2]*sqrt(m))))");
    pari("m=m++");
lb=pari("a");
D=pari("d");
CNT=pari("cnt");
for i in range(0,(CNT-1)):
    if D[i]%4==1:
        lb[i]=(1/2)*lb[i];
    else:
        D[i]=4*D[i];

• lb is the possible lower bound for the fundamental unit in the ring of integers of Q(sqrt(d))
  
  • a/b is the approximation of sqrt(d) from the first two terms of the continued fraction representation
  • lb=(a+b(sqrt(d)))/2 if d=1 mod 4
  • lb=a+b(sqrt(d)) if d=2 or 3 mod 4
    

point([(D[i],lb[i]) for i in range(0,(CNT-1))])

lower boud for the fundamental unit of the ring of integers for Q(sqrt(d)) vs. the discriminant D

Possible Resource: An Investigation of Bounds for the Regulator of Quadratic Fields (Jacobson, Lukes, Williams, Exp. Mathematics, Vol 4 (1995) No 3)

 

 

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

• FDN = list of # of prime factors of the discriminant for various negative values of d
• FDP = list of # of prime factors of the discriminant for various positive values of d
• P2hN = highest power of 2 dividing the class number for various negative values of d
• P2hP = highest power of 2 dividing the class number for various positive values of d

n=MP1+MP;
point([(FDP[i],P2hP[i]) for i in range(0,n)])

d > 0:  Highest power of 2 dividing the class number (y-axis) vs. Number of prime factors of the determinant (x-axis)

point([(FDN[i],P2hN[i]) for i in range(0,197)])

d < 0:  Highest power of 2 dividing the class number (y-axis) vs. Number of prime factors of the determinant (x-axis)