Copy_of_CL-small_cases

297 days ago by Ben.LeVeque

x=var('x') K.<a> = NumberField(x^2-x-1) 
       
E = EllipticCurve(K,[a,1,a+1,-3,-2*a]) 
       
E.conductor() 
        
Fractional ideal (6*a - 35)
Fractional ideal (6*a - 35)
E.discriminant() 
        
40*a + 93
40*a + 93
K.factor(E.discriminant()) 
        
(Fractional ideal (3*a - 1))^2 * (Fractional ideal (a + 9))
(Fractional ideal (3*a - 1))^2 * (Fractional ideal (a + 9))
F = E.short_weierstrass_model();F 
        
Elliptic Curve defined by y^2 = x^3 + (999*a-3942)*x + (-66960*a+75654)
over Number Field in a with defining polynomial x^2 - x - 1
Elliptic Curve defined by y^2 = x^3 + (999*a-3942)*x + (-66960*a+75654) over Number Field in a with defining polynomial x^2 - x - 1
F.discriminant() == 6^12*(-16*a+25) 
        
True
True
K.factor(F.discriminant()) 
        
(Fractional ideal (2))^12 * (Fractional ideal (3))^12 * (Fractional
ideal (5*a - 2))
(Fractional ideal (2))^12 * (Fractional ideal (3))^12 * (Fractional ideal (5*a - 2))
def f(delta): return EllipticCurve([0, 3^3*delta]) 
       
Ef = f(F.discriminant()) 
       
Ef.a_invariants() 
        
(0, 0, 0, 0, -940369969152*a + 1469328076800)
(0, 0, 0, 0, -940369969152*a + 1469328076800)
P = Ef.gens();P 
        
[(94608*a + 63504 : -44416512*a - 28926720 : 1)]
[(94608*a + 63504 : -44416512*a - 28926720 : 1)]
P[0] 
        
(94608*a + 63504 : -44416512*a - 28926720 : 1)
(94608*a + 63504 : -44416512*a - 28926720 : 1)
X,Y,_ = P[0];X,Y 
        
(94608*a + 63504, -44416512*a - 28926720)
(94608*a + 63504, -44416512*a - 28926720)
a_4 = X/(-12); a_6 = Y/108; a_4,a_6 
        
(-7884*a - 5292, -411264*a - 267840)
(-7884*a - 5292, -411264*a - 267840)
H = EllipticCurve(K,[a_4,a_6]) K.factor(H.discriminant()), '-------',K.factor(H.conductor()) 
        
((Fractional ideal (2))^12 * (Fractional ideal (3))^12 * (Fractional
ideal (5*a - 2)), '-------', (Fractional ideal (2))^5 * (Fractional
ideal (5*a - 2)))
((Fractional ideal (2))^12 * (Fractional ideal (3))^12 * (Fractional ideal (5*a - 2)), '-------', (Fractional ideal (2))^5 * (Fractional ideal (5*a - 2)))
K.factor(H.global_minimal_model().conductor()) 
        
(Fractional ideal (2))^5 * (Fractional ideal (5*a - 2))
(Fractional ideal (2))^5 * (Fractional ideal (5*a - 2))
H.conductor().norm() 
        
31744
31744
H.j_invariant() 
        
4660535296/31*a + 2880365888/31
4660535296/31*a + 2880365888/31
F.j_invariant() 
        
-106208/31*a + 51455/31
-106208/31*a + 51455/31
 
       
 
       
 
       
 
       
 
       
 
       
E.conductor() 
        
Fractional ideal (5*a - 2)
Fractional ideal (5*a - 2)
E = EllipticCurve([1,-1,0,-2,-1]);E 
        
Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 2*x - 1 over Rational
Field
Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 2*x - 1 over Rational Field
E.conductor() 
        
49
49
E.discriminant() 
        
-343
-343
F.is_isomorphic(E) 
        
True
True
F = E.short_weierstrass_model();F 
        
Elliptic Curve defined by y^2 = x^3 - 2835*x - 71442 over Rational Field
Elliptic Curve defined by y^2 = x^3 - 2835*x - 71442 over Rational Field
G = EllipticCurve([0,0,0,0,343*(64)^2*432^3]);G 
        
Elliptic Curve defined by y^2 = x^3 + 113267498287104 over Rational
Field
Elliptic Curve defined by y^2 = x^3 + 113267498287104 over Rational Field
int_list = G.integral_points();int_list 
        
[(-48384 : 0 : 1), (-30960 : 9142848 : 1), (16128 : 10838016 : 1),
(60480 : 18289152 : 1)]
[(-48384 : 0 : 1), (-30960 : 9142848 : 1), (16128 : 10838016 : 1), (60480 : 18289152 : 1)]
L=[] for i in int_list: L.append((i[0]/-2835, i[1]/-71442,1)) print L 
        
[(256/15, 0, 1), (688/63, -169312/1323, 1), (-256/45, -4096/27, 1),
(-64/3, -256, 1)]
[(256/15, 0, 1), (688/63, -169312/1323, 1), (-256/45, -4096/27, 1), (-64/3, -256, 1)]
for i in range(4): print (EllipticCurve([0,0,0,int_list[i][0]/(432*-64),int_list[i][1]/(432^2*64*2)]).conductor()) 
        
3136
117068856768
508032
243824
3136
117068856768
508032
243824
x1 = -2835 y1 = -71442 y1^2 -x1^3 + 113267498287104 
        
113295387779343
113295387779343
factor(432) 
        
2^4 * 3^3
2^4 * 3^3
E2 = EllipticCurve([1,0,1,-1,-2]) 
       
E2.short_weierstrass_model() 
        
Elliptic Curve defined by y^2 = x^3 - 675*x - 79650 over Rational Field
Elliptic Curve defined by y^2 = x^3 - 675*x - 79650 over Rational Field
E2.conductor() 
        
50
50
E2.discriminant() 
        
-1250
-1250
G2 = EllipticCurve([0,0,0,0,1250*(64)^2*432^3]);G2 
        
Elliptic Curve defined by y^2 = x^3 + 412782428160000 over Rational
Field
Elliptic Curve defined by y^2 = x^3 + 412782428160000 over Rational Field
G2.integral_points() 
        
[(14400 : 20390400 : 1), (172800 : 74649600 : 1)]
[(14400 : 20390400 : 1), (172800 : 74649600 : 1)]
14400/-675 
        
-64/3
-64/3
7464900/-79650 
        
-49766/531
-49766/531
x=var('x') K.<a> = NumberField(x^2-x-1) 
       
E1 = EllipticCurve(K,[0,0,0,-1,0]) 
       
E2 = EllipticCurve(K,[0,0,0,4,0]) 
       
E3 = EllipticCurve(K,[0,0,0,-11,-14]) 
       
E4 = EllipticCurve(K,[0,0,0,-11,14]) 
       
E1.is_isogenous(E4) 
        
True
True
E.conductor().norm() 
        
1024
1024
def ap(E,p): return E.change_ring(K.residue_field(p)).trace_of_frobenius() 
       
aplist=[] for p in [2,3,5,7,11,19,29,31,41,59,61,71,79,89]: for i in K.primes_above(p): try: aplist.append(ap(E3,i)) except: aplist.append('?') print aplist 
        
['?', -6, -2, -14, 0, 0, 0, 0, -10, -10, 0, 0, 10, 10, 0, 0, -10, -10,
0, 0, 0, 0, 10, 10]
['?', -6, -2, -14, 0, 0, 0, 0, -10, -10, 0, 0, 10, 10, 0, 0, -10, -10, 0, 0, 0, 0, 10, 10]
f_1 = open('/Users/bleveque/Sage/unmatched2000_729.txt') f_2 = open('/Users/bleveque/Sage/second_round_unmatched2000_no_last_unfound.txt') 
       
f = open('/Users/bleveque/Sage/matched_2000_8_1_11.txt') g = open('/Users/bleveque/Sage/matched_2000_8_1_11_currentlist.txt','w') 
       
counter=1 linenum = 1 for line in f.readlines(): print linenum linenum +=1 # try: if line.split('\n')[0].split(']')[1][0] == '[': g.write(line) counter+=1 # except IndexError: g.close() print counter 
        
WARNING: Output truncated!  
full_output.txt



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WARNING: Output truncated!  
full_output.txt



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f.readline() 
        
"1024 32
['?',-2,-2,6,0,-4,-4,-8,6,-2,8,-8,6,-10,8,-12,-2,14,0,8,-8,-8,10,-6]
unfound\n"
"1024 32 ['?',-2,-2,6,0,-4,-4,-8,6,-2,8,-8,6,-10,8,-12,-2,14,0,8,-8,-8,10,-6] unfound\n"
f.readline().split('\n')[0].split(']')[1][0] 
        
'['
'['