CoCalc -- 2021-11-25-2p1q5r5.ipynb

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ubuntu2204

Kernel: SageMath 10.0

In [1]:

#from sage.arith.misc import legendre_symbol
# sage.sage.rings.number_field import NumberField
#from sage.rings.all import NumberField
class color:
    BLUE = '\033[94m'
    YELLOW = '\033[93m'
    BOLD = '\033[1m'
    END = '\033[0m'

#from sage.all import *
from pprint import pprint
import json
preparser(True)

import sys

print(sys.version_info)

lista=[]
listb=[]

#Don't increase so much at once
#Let's check for 5000 First
#Then gradually increase
for i in range(1,50000):
    factoring = factor(sage_eval(str(i)))
    factors = str(factoring).split('*')
    #print(factors)
    if(len(factors) !=4):
        i=i
        #print("Not P^m*Q^n\n")
    else:
        factP = factors[0].split('^')
        factQ = factors[1].split('^')
        factR = factors[2].split('^')
        factS = factors[3].split('^')
        if(len(factP)>1 or len(factQ)>1 or len(factR)>1 or len(factS)>1):
            #print(factP)
            #print(factQ)
            #print("Not of the form  P^1*Q^1 \n")
            i=i
        else:
            if((int(factP[0])==2) and ((int(factQ[0])%8==1 and int(factR[0])%8==5 and int(factS[0])%8==5) or (int(factQ[0])%8==5 and int(factR[0])%8==1 and int(factS[0])%8==5) or (int(factQ[0])%8==5 and int(factR[0])%8==5 and int(factS[0])%8==1))):
                #print(i,int(factP[0]), int(factQ[0]))
                fact_str="n = "+str(i)+" p = "+str(int(factQ[0]))+" q = "+str(int(factR[0]))+" r = "+str(int(factS[0]))
                lspq=legendre_symbol(int(factQ[0]), int(factR[0]))
                lspr=legendre_symbol(int(factQ[0]), int(factS[0]))
                lsqr=legendre_symbol(int(factR[0]), int(factS[0]))
                lspq_str="(i) Legendre Symbol ((p/q) is: "+str(lspq)
                lspr_str="(ii) Legendre Symbol (p/r) is: "+str(lspr)
                lsqr_str="(iii) Legendre Symbol (q/r) is: "+str(lsqr)
                #print("(i) Legendre Symbol is: ", ls)
                #lista.insert(i,i)
                E = EllipticCurve([0,0,0,-i^2,0])
                E_str=str(E)
                #print(E)
                rank =  E.analytic_rank(algorithm='pari', leading_coefficient=False)
                #print("(iv) Rank is:",  rank)
                rank_str="(iv) Rank is:"+str(rank)
                K.<a> = NumberField(x^2+i/2)
                cn1 = K.class_number()
                #print("(v) Class Number of Q(sqrt(-n)) is:", cn)
                cn1_str="(v) Class Number of Q(sqrt(-n)) is:" + str(cn1)
                K.<a> = NumberField(x^2+i)
                cn2 = K.class_number()
                cn2_str="(vi) Class Number of Q(sqrt(-2n)) is:" + str(cn2)
                K.<a> = NumberField(x^2-i/2)
                cn3 = K.class_number()
                cn3_str="(vii) Class Number of Q(sqrt(n)) is:" + str(cn3)
                K.<a> = NumberField(x^2-i)
                cn4 = K.class_number()
                cn4_str="(viii) Class Number of Q(sqrt(2n)) is:" + str(cn4)
                end_str="-----------------"
                #print("(iv) Class Number of Q(sqrt(n)) is:", cn)
                #final_str = []
                if(rank!=0):
                    #lista.append(final_str)
                    lista.append(fact_str)
                    lista.append(lspq_str)
                    lista.append(lspr_str)
                    lista.append(E_str)
                    lista.append(rank_str)
                    lista.append(cn1_str)
                    lista.append(cn2_str)
                    lista.append(cn3_str)
                    lista.append(cn4_str) 
                    lista.append(end_str)
                elif(rank==0):
                    #listb.append(final_str)
                    listb.append(fact_str)
                    listb.append(lspq_str)
                    listb.append(lsqr_str)
                    listb.append(E_str)
                    listb.append(rank_str)
                    listb.append(cn1_str)
                    listb.append(cn2_str)
                    listb.append(cn3_str)
                    listb.append(cn4_str) 
                    listb.append(end_str)
                #print(E.sha())
                #try:
                    #ord = S.an(descent_second_limit=30)
                    #ord=S.p_primary_order(2)
                    #print("(v) 2-Primary Order is: ", ord)
                #except:
                    #print("Error in calculating 2-Primary Order")
                #print("----------------------")
print(color.BOLD + "FIRST CASE RANK: POSITIVE" + color.END)
print("-------------************************-------------")
#print(json.dumps(lista, indent=2))
#pprint(lista)
#print '\n'.join(lista)
print(*lista, sep='\n')
print("-------------************************-------------")
print("SECOND CASE RANK=0")
print("-------------************************-------------")
#print(json.dumps(listb, indent=2))
#pprint(listb)
print(*listb, sep='\n')
print("-------------************************-------------")
#pprint(listd)
#print(json.dumps(listd, indent=2))

#pprint("Rank 2, LS 1", lista)
#pprint("Rank 2, LS -1", listb)
#pprint("Rank 0, LS 1", listc)
#pprint("Rank 0, LS -1", listd)
#print("List B", listb)

Out[1]:

sys.version_info(major=3, minor=11, micro=1, releaselevel='final', serial=0)
FIRST CASE RANK: POSITIVE
-------------************************-------------
n = 6290 p = 5 q = 17 r = 37
(i) Legendre Symbol ((p/q) is: -1
(ii) Legendre Symbol (p/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 39564100*x over Rational Field
(iv) Rank is:2
(v) Class Number of Q(sqrt(-n)) is:32
(vi) Class Number of Q(sqrt(-2n)) is:48
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 9490 p = 5 q = 13 r = 73
(i) Legendre Symbol ((p/q) is: -1
(ii) Legendre Symbol (p/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 90060100*x over Rational Field
(iv) Rank is:2
(v) Class Number of Q(sqrt(-n)) is:80
(vi) Class Number of Q(sqrt(-2n)) is:64
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:16
-----------------
n = 10370 p = 5 q = 17 r = 61
(i) Legendre Symbol ((p/q) is: -1
(ii) Legendre Symbol (p/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 107536900*x over Rational Field
(iv) Rank is:2
(v) Class Number of Q(sqrt(-n)) is:32
(vi) Class Number of Q(sqrt(-2n)) is:144
(vii) Class Number of Q(sqrt(n)) is:20
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 15170 p = 5 q = 37 r = 41
(i) Legendre Symbol ((p/q) is: -1
(ii) Legendre Symbol (p/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 230128900*x over Rational Field
(iv) Rank is:2
(v) Class Number of Q(sqrt(-n)) is:32
(vi) Class Number of Q(sqrt(-2n)) is:96
(vii) Class Number of Q(sqrt(n)) is:8
(viii) Class Number of Q(sqrt(2n)) is:16
-----------------
n = 17810 p = 5 q = 13 r = 137
(i) Legendre Symbol ((p/q) is: -1
(ii) Legendre Symbol (p/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 317196100*x over Rational Field
(iv) Rank is:2
(v) Class Number of Q(sqrt(-n)) is:48
(vi) Class Number of Q(sqrt(-2n)) is:80
(vii) Class Number of Q(sqrt(n)) is:12
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 18530 p = 5 q = 17 r = 109
(i) Legendre Symbol ((p/q) is: -1
(ii) Legendre Symbol (p/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 343360900*x over Rational Field
(iv) Rank is:2
(v) Class Number of Q(sqrt(-n)) is:48
(vi) Class Number of Q(sqrt(-2n)) is:128
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:16
-----------------
n = 21170 p = 5 q = 29 r = 73
(i) Legendre Symbol ((p/q) is: 1
(ii) Legendre Symbol (p/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 448168900*x over Rational Field
(iv) Rank is:2
(v) Class Number of Q(sqrt(-n)) is:48
(vi) Class Number of Q(sqrt(-2n)) is:96
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:16
-----------------
n = 30914 p = 13 q = 29 r = 41
(i) Legendre Symbol ((p/q) is: 1
(ii) Legendre Symbol (p/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 955675396*x over Rational Field
(iv) Rank is:2
(v) Class Number of Q(sqrt(-n)) is:48
(vi) Class Number of Q(sqrt(-2n)) is:144
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 32770 p = 5 q = 29 r = 113
(i) Legendre Symbol ((p/q) is: 1
(ii) Legendre Symbol (p/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 1073872900*x over Rational Field
(iv) Rank is:2
(v) Class Number of Q(sqrt(-n)) is:128
(vi) Class Number of Q(sqrt(-2n)) is:96
(vii) Class Number of Q(sqrt(n)) is:12
(viii) Class Number of Q(sqrt(2n)) is:16
-----------------
n = 38690 p = 5 q = 53 r = 73
(i) Legendre Symbol ((p/q) is: -1
(ii) Legendre Symbol (p/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 1496916100*x over Rational Field
(iv) Rank is:2
(v) Class Number of Q(sqrt(-n)) is:64
(vi) Class Number of Q(sqrt(-2n)) is:144
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 41810 p = 5 q = 37 r = 113
(i) Legendre Symbol ((p/q) is: -1
(ii) Legendre Symbol (p/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 1748076100*x over Rational Field
(iv) Rank is:2
(v) Class Number of Q(sqrt(-n)) is:80
(vi) Class Number of Q(sqrt(-2n)) is:272
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 47090 p = 5 q = 17 r = 277
(i) Legendre Symbol ((p/q) is: -1
(ii) Legendre Symbol (p/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 2217468100*x over Rational Field
(iv) Rank is:2
(v) Class Number of Q(sqrt(-n)) is:48
(vi) Class Number of Q(sqrt(-2n)) is:224
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:32
-----------------
-------------************************-------------
SECOND CASE RANK=0
-------------************************-------------
n = 2210 p = 5 q = 13 r = 17
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 4884100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:16
(vi) Class Number of Q(sqrt(-2n)) is:56
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 4930 p = 5 q = 17 r = 29
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 24304900*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:32
(vi) Class Number of Q(sqrt(-2n)) is:32
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 5330 p = 5 q = 13 r = 41
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 28408900*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:32
(vi) Class Number of Q(sqrt(-2n)) is:104
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 9010 p = 5 q = 17 r = 53
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 81180100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:64
(vi) Class Number of Q(sqrt(-2n)) is:40
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 11570 p = 5 q = 13 r = 89
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 133864900*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:32
(vi) Class Number of Q(sqrt(-2n)) is:120
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 11890 p = 5 q = 29 r = 41
(i) Legendre Symbol ((p/q) is: 1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 141372100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:64
(vi) Class Number of Q(sqrt(-2n)) is:56
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 12610 p = 5 q = 13 r = 97
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 159012100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:96
(vi) Class Number of Q(sqrt(-2n)) is:64
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 12818 p = 13 q = 17 r = 29
(i) Legendre Symbol ((p/q) is: 1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 164301124*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:48
(vi) Class Number of Q(sqrt(-2n)) is:72
(vii) Class Number of Q(sqrt(n)) is:8
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 14690 p = 5 q = 13 r = 113
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 215796100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:32
(vi) Class Number of Q(sqrt(-2n)) is:120
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 16354 p = 13 q = 17 r = 37
(i) Legendre Symbol ((p/q) is: 1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 267453316*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:64
(vi) Class Number of Q(sqrt(-2n)) is:72
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:24
-----------------
n = 17170 p = 5 q = 17 r = 101
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 294808900*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:96
(vi) Class Number of Q(sqrt(-2n)) is:88
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 21730 p = 5 q = 41 r = 53
(i) Legendre Symbol ((p/q) is: 1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 472192900*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:96
(vi) Class Number of Q(sqrt(-2n)) is:56
(vii) Class Number of Q(sqrt(n)) is:12
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 23426 p = 13 q = 17 r = 53
(i) Legendre Symbol ((p/q) is: 1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 548777476*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:32
(vi) Class Number of Q(sqrt(-2n)) is:192
(vii) Class Number of Q(sqrt(n)) is:8
(viii) Class Number of Q(sqrt(2n)) is:16
-----------------
n = 25010 p = 5 q = 41 r = 61
(i) Legendre Symbol ((p/q) is: 1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 625500100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:64
(vi) Class Number of Q(sqrt(-2n)) is:144
(vii) Class Number of Q(sqrt(n)) is:16
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 25090 p = 5 q = 13 r = 193
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 629508100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:96
(vi) Class Number of Q(sqrt(-2n)) is:96
(vii) Class Number of Q(sqrt(n)) is:20
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 25330 p = 5 q = 17 r = 149
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 641608900*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:96
(vi) Class Number of Q(sqrt(-2n)) is:88
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 25810 p = 5 q = 29 r = 89
(i) Legendre Symbol ((p/q) is: 1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 666156100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:112
(vi) Class Number of Q(sqrt(-2n)) is:56
(vii) Class Number of Q(sqrt(n)) is:16
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 26690 p = 5 q = 17 r = 157
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 712356100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:80
(vi) Class Number of Q(sqrt(-2n)) is:200
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 26962 p = 13 q = 17 r = 61
(i) Legendre Symbol ((p/q) is: 1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 726949444*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:192
(vi) Class Number of Q(sqrt(-2n)) is:72
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 27010 p = 5 q = 37 r = 73
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 729540100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:96
(vi) Class Number of Q(sqrt(-2n)) is:56
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 28130 p = 5 q = 29 r = 97
(i) Legendre Symbol ((p/q) is: 1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 791296900*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:48
(vi) Class Number of Q(sqrt(-2n)) is:144
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:16
-----------------
n = 29410 p = 5 q = 17 r = 173
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 864948100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:128
(vi) Class Number of Q(sqrt(-2n)) is:96
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 30290 p = 5 q = 13 r = 233
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 917484100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:64
(vi) Class Number of Q(sqrt(-2n)) is:168
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 30770 p = 5 q = 17 r = 181
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 946792900*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:80
(vi) Class Number of Q(sqrt(-2n)) is:176
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:16
-----------------
n = 31330 p = 5 q = 13 r = 241
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 981568900*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:112
(vi) Class Number of Q(sqrt(-2n)) is:104
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:40
-----------------
n = 32930 p = 5 q = 37 r = 89
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 1084384900*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:48
(vi) Class Number of Q(sqrt(-2n)) is:200
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 33410 p = 5 q = 13 r = 257
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 1116228100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:80
(vi) Class Number of Q(sqrt(-2n)) is:200
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 33490 p = 5 q = 17 r = 197
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 1121580100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:80
(vi) Class Number of Q(sqrt(-2n)) is:80
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:48
-----------------
n = 35890 p = 5 q = 37 r = 97
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 1288092100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:80
(vi) Class Number of Q(sqrt(-2n)) is:112
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:16
-----------------
n = 36482 p = 17 q = 29 r = 37
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 1330936324*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:80
(vi) Class Number of Q(sqrt(-2n)) is:240
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:16
-----------------
n = 36530 p = 5 q = 13 r = 281
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 1334440900*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:48
(vi) Class Number of Q(sqrt(-2n)) is:168
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 38930 p = 5 q = 17 r = 229
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 1515544900*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:96
(vi) Class Number of Q(sqrt(-2n)) is:184
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 39442 p = 13 q = 37 r = 41
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 1555671364*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:144
(vi) Class Number of Q(sqrt(-2n)) is:72
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:40
-----------------
n = 39730 p = 5 q = 29 r = 137
(i) Legendre Symbol ((p/q) is: 1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 1578472900*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:208
(vi) Class Number of Q(sqrt(-2n)) is:96
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 40690 p = 5 q = 13 r = 313
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 1655676100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:192
(vi) Class Number of Q(sqrt(-2n)) is:104
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:24
-----------------
n = 41410 p = 5 q = 41 r = 101
(i) Legendre Symbol ((p/q) is: 1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 1714788100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:192
(vi) Class Number of Q(sqrt(-2n)) is:120
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 43810 p = 5 q = 13 r = 337
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 1919316100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:96
(vi) Class Number of Q(sqrt(-2n)) is:88
(vii) Class Number of Q(sqrt(n)) is:28
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 44530 p = 5 q = 61 r = 73
(i) Legendre Symbol ((p/q) is: 1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 1982920900*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:144
(vi) Class Number of Q(sqrt(-2n)) is:104
(vii) Class Number of Q(sqrt(n)) is:8
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 44642 p = 13 q = 17 r = 101
(i) Legendre Symbol ((p/q) is: 1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 1992908164*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:128
(vi) Class Number of Q(sqrt(-2n)) is:176
(vii) Class Number of Q(sqrt(n)) is:8
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 44690 p = 5 q = 41 r = 109
(i) Legendre Symbol ((p/q) is: 1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 1997196100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:48
(vi) Class Number of Q(sqrt(-2n)) is:184
(vii) Class Number of Q(sqrt(n)) is:8
(viii) Class Number of Q(sqrt(2n)) is:24
-----------------
n = 45730 p = 5 q = 17 r = 269
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 2091232900*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:192
(vi) Class Number of Q(sqrt(-2n)) is:96
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 45890 p = 5 q = 13 r = 353
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 2105892100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:96
(vi) Class Number of Q(sqrt(-2n)) is:176
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:16
-----------------
n = 47170 p = 5 q = 53 r = 89
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 2225008900*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:128
(vi) Class Number of Q(sqrt(-2n)) is:96
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:16
-----------------
n = 48178 p = 13 q = 17 r = 109
(i) Legendre Symbol ((p/q) is: 1
(iii) Legendre Symbol (q/r) is: -1
Elliptic Curve defined by y^2 = x^3 - 2321119684*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:128
(vi) Class Number of Q(sqrt(-2n)) is:72
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
n = 49810 p = 5 q = 17 r = 293
(i) Legendre Symbol ((p/q) is: -1
(iii) Legendre Symbol (q/r) is: 1
Elliptic Curve defined by y^2 = x^3 - 2481036100*x over Rational Field
(iv) Rank is:0
(v) Class Number of Q(sqrt(-n)) is:208
(vi) Class Number of Q(sqrt(-2n)) is:120
(vii) Class Number of Q(sqrt(n)) is:4
(viii) Class Number of Q(sqrt(2n)) is:8
-----------------
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