Fisher_Method_c++ -- Sage

open('/tmp/a.h','w').write(r""" #include "pari/pari.h" typedef long intPnt[3]; struct Elt { long a, b; }; Elt operator+(Elt x, Elt y) { Elt z; z.a = x.a + y.a; z.b = x.b + y.b; return z; } Elt operator*(Elt x, Elt y) { Elt z; long t; t = x.b*y.b; z.a = x.a*y.a + t; z.b = x.b*y.a + x.a*y.b + t; return z; } Elt disc,c4,c6; Elt xx,yy,zz,xy,xz,yz; Elt xxxz, xxyy, xyyz, yyyy, yyyz, xzzz, yyzz, zzzz; Elt Fcoeff[11], Gcoeff[11]; long ixxxx,ixyyy,ixxyy,iyyyy,ixxx,ixyy,iyyy,ixx,ixy,iyy; long intF0(long *coeff) { return coeff[1]*ixxyy+coeff[4]*iyyyy; } long intF1(long *coeff,long x,long y) { return coeff[0]*ixxx+coeff[1]*ixxyy+coeff[2]*ixx+coeff[3]*ixyy+coeff[4]*iyyyy+coeff[5]*ixy+coeff[6]*iyyy+coeff[7]*x+coeff[8]*iyy+coeff[9]*y+coeff[10]; } long intG0(long *coeff) { return coeff[0]*ixxxx+coeff[2]*ixxyy+coeff[4]*ixyyy+coeff[6]*iyyyy; } long intG1(long *coeff,long x,long y) { return coeff[0]*ixxxx+coeff[1]*ixxx+coeff[2]*ixxyy+coeff[3]*ixx*y+coeff[4]*ixyyy+coeff[5]*ixyy+coeff[6]*iyyyy+coeff[7]*iyyy+coeff[8]*x+coeff[9]*iyy+coeff[10]; } Elt F() { return Fcoeff[0]*xxxz + Fcoeff[1]*xxyy + Fcoeff[2]*xx*zz + Fcoeff[3]*xyyz + Fcoeff[4]*yyyy + Fcoeff[5]*xy*zz + Fcoeff[6]*yyyz + Fcoeff[7]*xzzz + Fcoeff[8]*yyzz + Fcoeff[9]*yz*zz + Fcoeff[10]*zzzz; } Elt G() { return Gcoeff[0]*xx*xx + Gcoeff[1]*xxxz + Gcoeff[2]*xxyy + Gcoeff[3]*xx*yz + Gcoeff[4]*xy*yy + Gcoeff[5]*xyyz + Gcoeff[6]*yyyy + Gcoeff[7]*yyyz + Gcoeff[8]*xzzz + Gcoeff[9]*yyzz + Gcoeff[10]*zzzz; } int naive_point_finder(Elt* x, Elt* y, Elt* z, long U) { Elt f,g; disc.a = -63; disc.b = 9; c4.a = 32; c4.b = 80; c6.a = -816; c6.b = -776; Fcoeff[0].a = 12; Fcoeff[0].b = 0; Fcoeff[1].a = 108; Fcoeff[1].b = 0; Fcoeff[2].a = 3; Fcoeff[2].b = 0; Fcoeff[2] = Fcoeff[2]*c4; Fcoeff[3].a = 72; Fcoeff[3].b = 0; Fcoeff[3] = Fcoeff[3]*c4; Fcoeff[4].a = -108; Fcoeff[4].b = 0; Fcoeff[4] = Fcoeff[4]*c4; Fcoeff[5].a = -12; Fcoeff[5].b = 0; Fcoeff[5] = Fcoeff[5]*c6; Fcoeff[6].a = 84; Fcoeff[6].b = 0; Fcoeff[6] = Fcoeff[6]*c6; Fcoeff[7] = c4*c4; Fcoeff[8].a = -15; Fcoeff[8].b = 0; Fcoeff[8] = Fcoeff[8]*Fcoeff[7]; Fcoeff[9] = c4*c6; Fcoeff[10].a = 768; Fcoeff[10].b = 0; Fcoeff[10] = Fcoeff[10]*disc; Gcoeff[0].a = 3; Gcoeff[0].b = 0; Gcoeff[1] = c4; Gcoeff[2].a = -18; Gcoeff[2].b = 0; Gcoeff[2] = Gcoeff[2]*c4; Gcoeff[3].a = -3; Gcoeff[3].b = 0; Gcoeff[3] = Gcoeff[3]*c6; Gcoeff[4].a = 24; Gcoeff[4].b = 0; Gcoeff[4] = Gcoeff[4]*c6; Gcoeff[5].a = 3; Gcoeff[5].b = 0; Gcoeff[5] = Gcoeff[5]*Fcoeff[7]; Gcoeff[6].a = -3; Gcoeff[6].b = 0; Gcoeff[6] = Gcoeff[6]*Gcoeff[5]; Gcoeff[7].a = -1; Gcoeff[7].b = 0; Gcoeff[7] = Gcoeff[7]*Fcoeff[9]; Gcoeff[8].a = 168; Gcoeff[8].b = 0; Gcoeff[8] = Gcoeff[8]*disc; Gcoeff[9].a = 1728; Gcoeff[9].b = 0; Gcoeff[9] = Gcoeff[9]*disc; Gcoeff[10].a = 5; Gcoeff[10].b = 0; Gcoeff[10] = Gcoeff[10]*disc*c4; long t,pU; t = U; long plist[10]; plist[0] = 11; plist[1] = 19; plist[2] = 29; plist[3] = 31; plist[4] = 41; plist[5] = 59; plist[6] = 61; plist[7] = 71; plist[8] = 79; plist[9] = 89; pU = 0; while(t) { t /= plist[pU]; pU += 1; } long res1, res2, i, j, k, p, r[2], fcoeff[11], gcoeff[11]; intPnt *fsol[2],*gsol[2]; long fmlen, gmlen; long fnumsol[2], gnumsol[2]; for(i=0;i<pU;i+=1) { p = plist[i]; r[0] = Fl_sqrt(5, p)+1; r[1] =((p+1)>>1)*r[0]%p; r[0] = p+1-r[1]; if(r[0] > r[1]) { r[1] = r[0]; r[0] = p+1-r[1]; } for(j=0;j<2;j++) { fmlen = p; gmlen = p; fsol[j] = (intPnt *)malloc(fmlen*sizeof(intPnt)); gsol[j] = (intPnt *)malloc(gmlen*sizeof(intPnt)); fnumsol[j] = 0; gnumsol[j] = 0; for(k=0;k<11;k++) { fcoeff[k] = (Fcoeff[k].a+Fcoeff[k].b*r[j])%p; fcoeff[k] += p*(fcoeff[k]<0); printf("%d\n",fcoeff[k]); gcoeff[k] = (Gcoeff[k].a+Gcoeff[k].b*r[j])%p; gcoeff[k] += p*(gcoeff[k]<0); } printf("\n"); for(res1=0;res1<p;res1++) { for(res2=0;res2<p;res2++) { ixx = res1*res1; ixy = res1*res2; iyy = res2*res2; ixxx = res1*ixx; ixyy = res1*iyy; iyyy = res2*iyy; ixxxx = ixx*ixx; ixyyy = ixy*iyy; ixxyy = ixy*ixy; iyyyy = iyy*iyy; if(!(intF0(fcoeff)%p)) { fsol[j][fnumsol[j]][0] = res1; fsol[j][fnumsol[j]][1] = res2; fsol[j][fnumsol[j]++][2] = 0; if(fnumsol[j]==fmlen) { fmlen <<= 1; fsol[j] = (intPnt *)realloc(fsol[j],fmlen*sizeof(intPnt)); } } if(!(intF1(fcoeff,res1,res2)%p)) { fsol[j][fnumsol[j]][0] = res1; fsol[j][fnumsol[j]][1] = res2; fsol[j][fnumsol[j]++][2] = 1; if(fnumsol[j]==fmlen) { fmlen <<= 1; fsol[j] = (intPnt *)realloc(fsol[j],fmlen*sizeof(intPnt)); } } if(!(intG0(gcoeff)%p)) { gsol[j][gnumsol[j]][0] = res1; gsol[j][gnumsol[j]][1] = res2; gsol[j][gnumsol[j]++][2] = 0; if(gnumsol[j]==gmlen) { gmlen <<= 1; gsol[j] = (intPnt *)realloc(gsol[j],gmlen*sizeof(intPnt)); } } if(!(intG1(gcoeff,res1,res2)%p)) { gsol[j][gnumsol[j]][0] = res1; gsol[j][gnumsol[j]][1] = res2; gsol[j][gnumsol[j]++][2] = 1; if(gnumsol[j]==gmlen) { gmlen <<= 1; gsol[j] = (intPnt *)realloc(gsol[j],gmlen*sizeof(intPnt)); } } } } } printf("%d %d\n",fnumsol[0]*fnumsol[1],gnumsol[0]*gnumsol[1]); free(fsol[0]); free(fsol[1]); free(gsol[0]); free(gsol[1]); } return 0; for(x->a=-U; x->a<=U; x->a++){ for(x->b=-U; x->b<=U; x->b++){ for(y->a=-U; y->a<=U; y->a++){ for(y->b=-U; y->b<=U; y->b++){ for(z->a=-U; z->a<=U; z->a++){ if(z->a != 0) { xx = (*x)*(*x); yy = (*y)*(*y); zz = (*z)*(*z); xy = (*x)*(*y); xz = (*x)*(*z); yz = (*y)*(*z); xxxz = xx*xz; xxyy = xx*yy; xyyz = xy*yz; yyyy = yy*yy; yyyz = yy*yz; xzzz = xz*zz; yyzz = yy*zz; zzzz = zz*zz; f = F(); if(f.a == 0 && f.b == 0) { printf("F\n"); return 0; } g = G(); if(g.a == 0 && g.b == 0) { printf("G\n"); return 0; } } } } } } } return 1; } """)

%cython #clang c++ from sage.rings.integer import Integer from psage.number_fields.sqrt5.misc import a cdef extern from "/tmp/a.h": cdef cppclass Elt: long a, b int naive_point_finder(Elt *x, Elt *y, Elt *z, long U) def point_finder_wrapper(long B): cdef Elt x,y,z if naive_point_finder(&x, &y, &z, B): return None return [x.a+x.b*a,y.a+y.b*a,z.a+z.b*a]

__home_oha...ode_sage159_spyx.cpp __home_oha...de_sage159_spyx.html

point_finder_wrapper(30)

1
9
0
0
0
4
5
0
0
0
10

1
9
3
6
2
6
2
1
7
5
4

1400 455
12
13
4
1
8
17
14
6
5
15
8

12
13
10
12
1
18
7
9
17
14
6

2736 798
[140653129109104*a + 99082324, 99387696*a, 6537344*a + 77972904]

1
9
0
0
0
4
5
0
0
0
10

1
9
3
6
2
6
2
1
7
5
4

1400 455
12
13
4
1
8
17
14
6
5
15
8

12
13
10
12
1
18
7
9
17
14
6

2736 798
[140653129109104*a + 99082324, 99387696*a, 6537344*a + 77972904]

len(mod19list)

FR = lambda R,x,y,z: R(12)*x^3*z + R(108)*x^2*y^2 + R(3*c4)*x^2*z^2 + R(72*c4)*x*y^2*z + R(-108*c4)*y^4 + R(-12*c6)*x*y*z^2 + R(84*c6)*y^3*z + R(c4^2)*x*z^3 + R(-15*c4^2)*y^2*z^2 + R(c4*c6)*y*z^3 + R(768*disc)*z^4 GR = lambda R,x,y,z: R(3)*x^4 + R(c4)*x^3*z + R(-18*c4)*x^2*y^2 + R(-3*c6)*x^2*y*z + R(24*c6)*x*y^3 + R(3*c4^2)*x*y^2*z + R(-9*c4^2)*y^4 + R(-c4*c6)*y^3*z + R(168*disc)*x*z^3 + R(1728*disc)*y^2*z^2 + R(5*c4*disc)*z^4

FR(R,var('x'),var('y'),var('z'))

x^3*z + 9*x^2*y^2 + 4*x*y*z^2 + 5*y^3*z + 10*z^4

x^3*z + 9*x^2*y^2 + 4*x*y*z^2 + 5*y^3*z + 10*z^4

Llist = [(x,y,1) for x,y in solve_mod(GR(L,var('x'),var('y'),1)==0,11)] + [(x,y,0) for x,y in solve_mod(GR(L,var('x'),var('y'),0)==0,11)] Rlist = [(x,y,1) for x,y in solve_mod(GR(R,var('x'),var('y'),1)==0,11)] + [(x,y,0) for x,y in solve_mod(GR(R,var('x'),var('y'),0)==0,11)]

len(Llist)

13*35

len(Rlist)

type(a)

<type
'sage.rings.number_field.number_field_element_quadratic.NumberFieldEleme\
nt_quadratic'>

<type 'sage.rings.number_field.number_field_element_quadratic.NumberFieldElement_quadratic'>

1400

for t in mod11list: t[2].norm()

WARNING: Output truncated!  
full_output.txt



1
1
1
1
1
1
1
1
1
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
1
1
1
1
1
1
1
1
1
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44

...

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
121
121
121
121
121
121
121
121
121
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

WARNING: Output truncated!  
full_output.txt



1
1
1
1
1
1
1
1
1
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44
1
1
1
1
1
1
1
1
1
-44
-44
-44
-44
-44
-44
-44
-44
-44
-44

...

0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
121
121
121
121
121
121
121
121
121
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

full_output.txt

Rlist

[(0, 10, 1), (0, 12, 1), (1, 12, 1), (1, 13, 1), (7, 8, 1), (8, 10, 1),
(8, 18, 1), (10, 10, 1), (11, 0, 1), (11, 16, 1), (12, 1, 1), (12, 11,
1), (13, 8, 1), (13, 17, 1), (16, 3, 1), (16, 17, 1), (17, 14, 1), (17,
17, 1), (18, 8, 1), (0, 0, 0), (1, 0, 0), (2, 0, 0), (3, 0, 0), (4, 0,
0), (5, 0, 0), (6, 0, 0), (7, 0, 0), (8, 0, 0), (9, 0, 0), (10, 0, 0),
(11, 0, 0), (12, 0, 0), (13, 0, 0), (14, 0, 0), (15, 0, 0), (16, 0, 0),
(17, 0, 0), (18, 0, 0)]

[(0, 10, 1), (0, 12, 1), (1, 12, 1), (1, 13, 1), (7, 8, 1), (8, 10, 1), (8, 18, 1), (10, 10, 1), (11, 0, 1), (11, 16, 1), (12, 1, 1), (12, 11, 1), (13, 8, 1), (13, 17, 1), (16, 3, 1), (16, 17, 1), (17, 14, 1), (17, 17, 1), (18, 8, 1), (0, 0, 0), (1, 0, 0), (2, 0, 0), (3, 0, 0), (4, 0, 0), (5, 0, 0), (6, 0, 0), (7, 0, 0), (8, 0, 0), (9, 0, 0), (10, 0, 0), (11, 0, 0), (12, 0, 0), (13, 0, 0), (14, 0, 0), (15, 0, 0), (16, 0, 0), (17, 0, 0), (18, 0, 0)]

mod11list = mod11list[:1400]

len(mod11list)*len(mod19list)

(11*19)^6

83344647990241

83344647990241

83344647990241

mod19list = [] for s in Rlist: for t in Llist: b = ((t[i],(t[i]-s[i])*2%19) for i in range(3)) mod19list.append(tuple((Integer((c-d*15)%19)+Integer(d)*a for c,d in b))) mod19list

WARNING: Output truncated!  
full_output.txt



[(0, 4*a + 9, 1), (6*a + 8, 5*a + 4, 1), (6*a + 8, 15*a + 11, 1), (12*a
+ 16, 6*a + 18, 1), (18*a + 5, 11*a + 12, 1), (18*a + 5, 10, 1), (18*a +
5, 14*a + 16, 1), (a + 14, 3*a + 14, 1), (3*a + 4, 16*a + 6, 1), (5*a +
13, 17*a + 1, 1), (7*a + 3, 18*a + 15, 1), (7*a + 3, 5*a + 4, 1), (7*a +
3, 8*a + 8, 1), (9*a + 12, 2*a, 1), (11*a + 2, 9*a + 3, 1), (11*a + 2,
2*a, 1), (17*a + 10, 2*a, 1), (0, 18*a + 15, 17*a + 11), (2*a + 9, 18*a
+ 15, 17*a + 11), (2*a + 9, 9*a + 3, 17*a + 11), (2*a + 9, 8*a + 8, 17*a
+ 11), (4*a + 18, 18*a + 15, 17*a + 11), (4*a + 18, 17*a + 1, 17*a +
11), (4*a + 18, 10, 17*a + 11), (6*a + 8, 18*a + 15, 17*a + 11), (6*a +
8, 7*a + 13, 17*a + 11), (6*a + 8, 10*a + 17, 17*a + 11), (8*a + 17,
18*a + 15, 17*a + 11), (8*a + 17, a + 5, 17*a + 11), (8*a + 17, 16*a +
6, 17*a + 11), (10*a + 7, 18*a + 15, 17*a + 11), (10*a + 7, 11*a + 12,
17*a + 11), (10*a + 7, 6*a + 18, 17*a + 11), (12*a + 16, 18*a + 15, 17*a
+ 11), (12*a + 16, 15*a + 11, 17*a + 11), (12*a + 16, 2*a, 17*a + 11),
(14*a + 6, 18*a + 15, 17*a + 11), (14*a + 6, 5*a + 4, 17*a + 11), (14*a
+ 6, 12*a + 7, 17*a + 11), (16*a + 15, 18*a + 15, 17*a + 11), (16*a +
15, 3*a + 14, 17*a + 11), (16*a + 15, 14*a + 16, 17*a + 11), (18*a + 5,
18*a + 15, 17*a + 11), (18*a + 5, 13*a + 2, 17*a + 11), (18*a + 5, 4*a +
9, 17*a + 11), (a + 14, 18*a + 15, 17*a + 11), (a + 14, 13*a + 2, 17*a +
11), (a + 14, 4*a + 9, 17*a + 11), (3*a + 4, 18*a + 15, 17*a + 11), (3*a
+ 4, 3*a + 14, 17*a + 11), (3*a + 4, 14*a + 16, 17*a + 11), (5*a + 13,
18*a + 15, 17*a + 11), (5*a + 13, 5*a + 4, 17*a + 11), (5*a + 13, 12*a +
7, 17*a + 11), (7*a + 3, 18*a + 15, 17*a + 11), (7*a + 3, 15*a + 11,
17*a + 11), (7*a + 3, 2*a, 17*a + 11), (9*a + 12, 18*a + 15, 17*a + 11),
(9*a + 12, 11*a + 12, 17*a + 11), (9*a + 12, 6*a + 18, 17*a + 11), (11*a
+ 2, 18*a + 15, 17*a + 11), (11*a + 2, a + 5, 17*a + 11), (11*a + 2,
16*a + 6, 17*a + 11), (13*a + 11, 18*a + 15, 17*a + 11), (13*a + 11, 7*a
+ 13, 17*a + 11), (13*a + 11, 10*a + 17, 17*a + 11), (15*a + 1, 18*a +
15, 17*a + 11), (15*a + 1, 17*a + 1, 17*a + 11), (15*a + 1, 10, 17*a +
11), (17*a + 10, 18*a + 15, 17*a + 11), (17*a + 10, 9*a + 3, 17*a + 11),
(17*a + 10, 8*a + 8, 17*a + 11), (0, 12, 1), (6*a + 8, a + 7, 1), (6*a +
8, 11*a + 14, 1), (12*a + 16, 2*a + 2, 1), (18*a + 5, 7*a + 15, 1),
(18*a + 5, 15*a + 13, 1), (18*a + 5, 10*a, 1), (a + 14, 18*a + 17, 1),
(3*a + 4, 12*a + 9, 1), (5*a + 13, 13*a + 4, 1), (7*a + 3, 14*a + 18,
1), (7*a + 3, a + 7, 1), (7*a + 3, 4*a + 11, 1), (9*a + 12, 17*a + 3,
1), (11*a + 2, 5*a + 6, 1), (11*a + 2, 17*a + 3, 1), (17*a + 10, 17*a +
3, 1), (0, 14*a + 18, 17*a + 11), (2*a + 9, 14*a + 18, 17*a + 11), (2*a
+ 9, 5*a + 6, 17*a + 11), (2*a + 9, 4*a + 11, 17*a + 11), (4*a + 18,
14*a + 18, 17*a + 11), (4*a + 18, 13*a + 4, 17*a + 11), (4*a + 18, 15*a
+ 13, 17*a + 11), (6*a + 8, 14*a + 18, 17*a + 11), (6*a + 8, 3*a + 16,
17*a + 11), (6*a + 8, 6*a + 1, 17*a + 11), (8*a + 17, 14*a + 18, 17*a +
11), (8*a + 17, 16*a + 8, 17*a + 11), (8*a + 17, 12*a + 9, 17*a + 11),
(10*a + 7, 14*a + 18, 17*a + 11), (10*a + 7, 7*a + 15, 17*a + 11), (10*a
+ 7, 2*a + 2, 17*a + 11), (12*a + 16, 14*a + 18, 17*a + 11), (12*a + 16,
11*a + 14, 17*a + 11), (12*a + 16, 17*a + 3, 17*a + 11), (14*a + 6, 14*a
+ 18, 17*a + 11), (14*a + 6, a + 7, 17*a + 11), (14*a + 6, 8*a + 10,
17*a + 11), (16*a + 15, 14*a + 18, 17*a + 11), (16*a + 15, 18*a + 17,
17*a + 11), (16*a + 15, 10*a, 17*a + 11), (18*a + 5, 14*a + 18, 17*a +
11), (18*a + 5, 9*a + 5, 17*a + 11), (18*a + 5, 12, 17*a + 11), (a + 14,
14*a + 18, 17*a + 11), (a + 14, 9*a + 5, 17*a + 11), (a + 14, 12, 17*a +
11), (3*a + 4, 14*a + 18, 17*a + 11), (3*a + 4, 18*a + 17, 17*a + 11),
(3*a + 4, 10*a, 17*a + 11), (5*a + 13, 14*a + 18, 17*a + 11), (5*a + 13,
a + 7, 17*a + 11), (5*a + 13, 8*a + 10, 17*a + 11), (7*a + 3, 14*a + 18,
17*a + 11), (7*a + 3, 11*a + 14, 17*a + 11), (7*a + 3, 17*a + 3, 17*a +
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9), (16*a + 6, 3*a + 4, 2*a + 9), (18*a + 15, 0, 0), (a + 5, 0, 0), (a +
5, 10*a + 7, 0), (a + 5, 9*a + 12, 0), (3*a + 14, 0, 0), (3*a + 14, 18*a
+ 5, 0), (3*a + 14, a + 14, 0), (5*a + 4, 0, 0), (5*a + 4, 8*a + 17, 0),
(5*a + 4, 11*a + 2, 0), (7*a + 13, 0, 0), (7*a + 13, 2*a + 9, 0), (7*a +
13, 17*a + 10, 0), (9*a + 3, 0, 0), (9*a + 3, 12*a + 16, 0), (9*a + 3,
7*a + 3, 0), (11*a + 12, 0, 0), (11*a + 12, 16*a + 15, 0), (11*a + 12,
3*a + 4, 0), (13*a + 2, 0, 0), (13*a + 2, 6*a + 8, 0), (13*a + 2, 13*a +
11, 0), (15*a + 11, 0, 0), (15*a + 11, 4*a + 18, 0), (15*a + 11, 15*a +
1, 0), (17*a + 1, 0, 0), (17*a + 1, 14*a + 6, 0), (17*a + 1, 5*a + 13,
0), (10, 0, 0), (10, 14*a + 6, 0), (10, 5*a + 13, 0), (2*a, 0, 0), (2*a,
4*a + 18, 0), (2*a, 15*a + 1, 0), (4*a + 9, 0, 0), (4*a + 9, 6*a + 8,
0), (4*a + 9, 13*a + 11, 0), (6*a + 18, 0, 0), (6*a + 18, 16*a + 15, 0),
(6*a + 18, 3*a + 4, 0), (8*a + 8, 0, 0), (8*a + 8, 12*a + 16, 0), (8*a +
8, 7*a + 3, 0), (10*a + 17, 0, 0), (10*a + 17, 2*a + 9, 0), (10*a + 17,
17*a + 10, 0), (12*a + 7, 0, 0), (12*a + 7, 8*a + 17, 0), (12*a + 7,
11*a + 2, 0), (14*a + 16, 0, 0), (14*a + 16, 18*a + 5, 0), (14*a + 16, a
+ 14, 0), (16*a + 6, 0, 0), (16*a + 6, 10*a + 7, 0), (16*a + 6, 9*a +
12, 0), (16*a + 7, 5*a + 13, 2*a + 9), (3*a + 15, 6*a + 8, 2*a + 9),
(3*a + 15, 16*a + 15, 2*a + 9), (9*a + 4, 7*a + 3, 2*a + 9), (15*a + 12,
12*a + 16, 2*a + 9), (15*a + 12, a + 14, 2*a + 9), (15*a + 12, 15*a + 1,
2*a + 9), (17*a + 2, 4*a + 18, 2*a + 9), (11, 17*a + 10, 2*a + 9), (2*a
+ 1, 18*a + 5, 2*a + 9), (4*a + 10, 0, 2*a + 9), (4*a + 10, 6*a + 8, 2*a
+ 9), (4*a + 10, 9*a + 12, 2*a + 9), (6*a, 3*a + 4, 2*a + 9), (8*a + 9,
10*a + 7, 2*a + 9), (8*a + 9, 3*a + 4, 2*a + 9), (14*a + 17, 3*a + 4,
2*a + 9), (16*a + 7, 0, 0), (18*a + 16, 0, 0), (18*a + 16, 10*a + 7, 0),
(18*a + 16, 9*a + 12, 0), (a + 6, 0, 0), (a + 6, 18*a + 5, 0), (a + 6, a
+ 14, 0), (3*a + 15, 0, 0), (3*a + 15, 8*a + 17, 0), (3*a + 15, 11*a +
2, 0), (5*a + 5, 0, 0), (5*a + 5, 2*a + 9, 0), (5*a + 5, 17*a + 10, 0),
(7*a + 14, 0, 0), (7*a + 14, 12*a + 16, 0), (7*a + 14, 7*a + 3, 0), (9*a
+ 4, 0, 0), (9*a + 4, 16*a + 15, 0), (9*a + 4, 3*a + 4, 0), (11*a + 13,
0, 0), (11*a + 13, 6*a + 8, 0), (11*a + 13, 13*a + 11, 0), (13*a + 3, 0,
0), (13*a + 3, 4*a + 18, 0), (13*a + 3, 15*a + 1, 0), (15*a + 12, 0, 0),
(15*a + 12, 14*a + 6, 0), (15*a + 12, 5*a + 13, 0), (17*a + 2, 0, 0),
(17*a + 2, 14*a + 6, 0), (17*a + 2, 5*a + 13, 0), (11, 0, 0), (11, 4*a +
18, 0), (11, 15*a + 1, 0), (2*a + 1, 0, 0), (2*a + 1, 6*a + 8, 0), (2*a
+ 1, 13*a + 11, 0), (4*a + 10, 0, 0), (4*a + 10, 16*a + 15, 0), (4*a +
10, 3*a + 4, 0), (6*a, 0, 0), (6*a, 12*a + 16, 0), (6*a, 7*a + 3, 0),
(8*a + 9, 0, 0), (8*a + 9, 2*a + 9, 0), (8*a + 9, 17*a + 10, 0), (10*a +
18, 0, 0), (10*a + 18, 8*a + 17, 0), (10*a + 18, 11*a + 2, 0), (12*a +
8, 0, 0), (12*a + 8, 18*a + 5, 0), (12*a + 8, a + 14, 0), (14*a + 17, 0,
0), (14*a + 17, 10*a + 7, 0), (14*a + 17, 9*a + 12, 0), (14*a + 18, 5*a
+ 13, 2*a + 9), (a + 7, 6*a + 8, 2*a + 9), (a + 7, 16*a + 15, 2*a + 9),
(7*a + 15, 7*a + 3, 2*a + 9), (13*a + 4, 12*a + 16, 2*a + 9), (13*a + 4,
a + 14, 2*a + 9), (13*a + 4, 15*a + 1, 2*a + 9), (15*a + 13, 4*a + 18,
2*a + 9), (17*a + 3, 17*a + 10, 2*a + 9), (12, 18*a + 5, 2*a + 9), (2*a
+ 2, 0, 2*a + 9), (2*a + 2, 6*a + 8, 2*a + 9), (2*a + 2, 9*a + 12, 2*a +
9), (4*a + 11, 3*a + 4, 2*a + 9), (6*a + 1, 10*a + 7, 2*a + 9), (6*a +
1, 3*a + 4, 2*a + 9), (12*a + 9, 3*a + 4, 2*a + 9), (14*a + 18, 0, 0),
(16*a + 8, 0, 0), (16*a + 8, 10*a + 7, 0), (16*a + 8, 9*a + 12, 0),
(18*a + 17, 0, 0), (18*a + 17, 18*a + 5, 0), (18*a + 17, a + 14, 0), (a
+ 7, 0, 0), (a + 7, 8*a + 17, 0), (a + 7, 11*a + 2, 0), (3*a + 16, 0,
0), (3*a + 16, 2*a + 9, 0), (3*a + 16, 17*a + 10, 0), (5*a + 6, 0, 0),
(5*a + 6, 12*a + 16, 0), (5*a + 6, 7*a + 3, 0), (7*a + 15, 0, 0), (7*a +
15, 16*a + 15, 0), (7*a + 15, 3*a + 4, 0), (9*a + 5, 0, 0), (9*a + 5,
6*a + 8, 0), (9*a + 5, 13*a + 11, 0), (11*a + 14, 0, 0), (11*a + 14, 4*a
+ 18, 0), (11*a + 14, 15*a + 1, 0), (13*a + 4, 0, 0), (13*a + 4, 14*a +
6, 0), (13*a + 4, 5*a + 13, 0), (15*a + 13, 0, 0), (15*a + 13, 14*a + 6,
0), (15*a + 13, 5*a + 13, 0), (17*a + 3, 0, 0), (17*a + 3, 4*a + 18, 0),
(17*a + 3, 15*a + 1, 0), (12, 0, 0), (12, 6*a + 8, 0), (12, 13*a + 11,
0), (2*a + 2, 0, 0), (2*a + 2, 16*a + 15, 0), (2*a + 2, 3*a + 4, 0),
(4*a + 11, 0, 0), (4*a + 11, 12*a + 16, 0), (4*a + 11, 7*a + 3, 0), (6*a
+ 1, 0, 0), (6*a + 1, 2*a + 9, 0), (6*a + 1, 17*a + 10, 0), (8*a + 10,
0, 0), (8*a + 10, 8*a + 17, 0), (8*a + 10, 11*a + 2, 0), (10*a, 0, 0),
(10*a, 18*a + 5, 0), (10*a, a + 14, 0), (12*a + 9, 0, 0), (12*a + 9,
10*a + 7, 0), (12*a + 9, 9*a + 12, 0), (12*a + 10, 5*a + 13, 2*a + 9),
(18*a + 18, 6*a + 8, 2*a + 9), (18*a + 18, 16*a + 15, 2*a + 9), (5*a +
7, 7*a + 3, 2*a + 9), (11*a + 15, 12*a + 16, 2*a + 9), (11*a + 15, a +
14, 2*a + 9), (11*a + 15, 15*a + 1, 2*a + 9), (13*a + 5, 4*a + 18, 2*a +
9), (15*a + 14, 17*a + 10, 2*a + 9), (17*a + 4, 18*a + 5, 2*a + 9), (13,
0, 2*a + 9), (13, 6*a + 8, 2*a + 9), (13, 9*a + 12, 2*a + 9), (2*a + 3,
3*a + 4, 2*a + 9), (4*a + 12, 10*a + 7, 2*a + 9), (4*a + 12, 3*a + 4,
2*a + 9), (10*a + 1, 3*a + 4, 2*a + 9), (12*a + 10, 0, 0), (14*a, 0, 0),
(14*a, 10*a + 7, 0), (14*a, 9*a + 12, 0), (16*a + 9, 0, 0), (16*a + 9,
18*a + 5, 0), (16*a + 9, a + 14, 0), (18*a + 18, 0, 0), (18*a + 18, 8*a
+ 17, 0), (18*a + 18, 11*a + 2, 0), (a + 8, 0, 0), (a + 8, 2*a + 9, 0),
(a + 8, 17*a + 10, 0), (3*a + 17, 0, 0), (3*a + 17, 12*a + 16, 0), (3*a
+ 17, 7*a + 3, 0), (5*a + 7, 0, 0), (5*a + 7, 16*a + 15, 0), (5*a + 7,
3*a + 4, 0), (7*a + 16, 0, 0), (7*a + 16, 6*a + 8, 0), (7*a + 16, 13*a +
11, 0), (9*a + 6, 0, 0), (9*a + 6, 4*a + 18, 0), (9*a + 6, 15*a + 1, 0),
(11*a + 15, 0, 0), (11*a + 15, 14*a + 6, 0), (11*a + 15, 5*a + 13, 0),
(13*a + 5, 0, 0), (13*a + 5, 14*a + 6, 0), (13*a + 5, 5*a + 13, 0),
(15*a + 14, 0, 0), (15*a + 14, 4*a + 18, 0), (15*a + 14, 15*a + 1, 0),
(17*a + 4, 0, 0), (17*a + 4, 6*a + 8, 0), (17*a + 4, 13*a + 11, 0), (13,
0, 0), (13, 16*a + 15, 0), (13, 3*a + 4, 0), (2*a + 3, 0, 0), (2*a + 3,
12*a + 16, 0), (2*a + 3, 7*a + 3, 0), (4*a + 12, 0, 0), (4*a + 12, 2*a +
9, 0), (4*a + 12, 17*a + 10, 0), (6*a + 2, 0, 0), (6*a + 2, 8*a + 17,
0), (6*a + 2, 11*a + 2, 0), (8*a + 11, 0, 0), (8*a + 11, 18*a + 5, 0),
(8*a + 11, a + 14, 0), (10*a + 1, 0, 0), (10*a + 1, 10*a + 7, 0), (10*a
+ 1, 9*a + 12, 0), (10*a + 2, 5*a + 13, 2*a + 9), (16*a + 10, 6*a + 8,
2*a + 9), (16*a + 10, 16*a + 15, 2*a + 9), (3*a + 18, 7*a + 3, 2*a + 9),
(9*a + 7, 12*a + 16, 2*a + 9), (9*a + 7, a + 14, 2*a + 9), (9*a + 7,
15*a + 1, 2*a + 9), (11*a + 16, 4*a + 18, 2*a + 9), (13*a + 6, 17*a +
10, 2*a + 9), (15*a + 15, 18*a + 5, 2*a + 9), (17*a + 5, 0, 2*a + 9),
(17*a + 5, 6*a + 8, 2*a + 9), (17*a + 5, 9*a + 12, 2*a + 9), (14, 3*a +
4, 2*a + 9), (2*a + 4, 10*a + 7, 2*a + 9), (2*a + 4, 3*a + 4, 2*a + 9),
(8*a + 12, 3*a + 4, 2*a + 9), (10*a + 2, 0, 0), (12*a + 11, 0, 0), (12*a
+ 11, 10*a + 7, 0), (12*a + 11, 9*a + 12, 0), (14*a + 1, 0, 0), (14*a +
1, 18*a + 5, 0), (14*a + 1, a + 14, 0), (16*a + 10, 0, 0), (16*a + 10,
8*a + 17, 0), (16*a + 10, 11*a + 2, 0), (18*a, 0, 0), (18*a, 2*a + 9,
0), (18*a, 17*a + 10, 0), (a + 9, 0, 0), (a + 9, 12*a + 16, 0), (a + 9,
7*a + 3, 0), (3*a + 18, 0, 0), (3*a + 18, 16*a + 15, 0), (3*a + 18, 3*a
+ 4, 0), (5*a + 8, 0, 0), (5*a + 8, 6*a + 8, 0), (5*a + 8, 13*a + 11,
0), (7*a + 17, 0, 0), (7*a + 17, 4*a + 18, 0), (7*a + 17, 15*a + 1, 0),
(9*a + 7, 0, 0), (9*a + 7, 14*a + 6, 0), (9*a + 7, 5*a + 13, 0), (11*a +
16, 0, 0), (11*a + 16, 14*a + 6, 0), (11*a + 16, 5*a + 13, 0), (13*a +
6, 0, 0), (13*a + 6, 4*a + 18, 0), (13*a + 6, 15*a + 1, 0), (15*a + 15,
0, 0), (15*a + 15, 6*a + 8, 0), (15*a + 15, 13*a + 11, 0), (17*a + 5, 0,
0), (17*a + 5, 16*a + 15, 0), (17*a + 5, 3*a + 4, 0), (14, 0, 0), (14,
12*a + 16, 0), (14, 7*a + 3, 0), (2*a + 4, 0, 0), (2*a + 4, 2*a + 9, 0),
(2*a + 4, 17*a + 10, 0), (4*a + 13, 0, 0), (4*a + 13, 8*a + 17, 0), (4*a
+ 13, 11*a + 2, 0), (6*a + 3, 0, 0), (6*a + 3, 18*a + 5, 0), (6*a + 3, a
+ 14, 0), (8*a + 12, 0, 0), (8*a + 12, 10*a + 7, 0), (8*a + 12, 9*a +
12, 0), (8*a + 13, 5*a + 13, 2*a + 9), (14*a + 2, 6*a + 8, 2*a + 9),
(14*a + 2, 16*a + 15, 2*a + 9), (a + 10, 7*a + 3, 2*a + 9), (7*a + 18,
12*a + 16, 2*a + 9), (7*a + 18, a + 14, 2*a + 9), (7*a + 18, 15*a + 1,
2*a + 9), (9*a + 8, 4*a + 18, 2*a + 9), (11*a + 17, 17*a + 10, 2*a + 9),
(13*a + 7, 18*a + 5, 2*a + 9), (15*a + 16, 0, 2*a + 9), (15*a + 16, 6*a
+ 8, 2*a + 9), (15*a + 16, 9*a + 12, 2*a + 9), (17*a + 6, 3*a + 4, 2*a +
9), (15, 10*a + 7, 2*a + 9), (15, 3*a + 4, 2*a + 9), (6*a + 4, 3*a + 4,
2*a + 9), (8*a + 13, 0, 0), (10*a + 3, 0, 0), (10*a + 3, 10*a + 7, 0),
(10*a + 3, 9*a + 12, 0), (12*a + 12, 0, 0), (12*a + 12, 18*a + 5, 0),
(12*a + 12, a + 14, 0), (14*a + 2, 0, 0), (14*a + 2, 8*a + 17, 0), (14*a
+ 2, 11*a + 2, 0), (16*a + 11, 0, 0), (16*a + 11, 2*a + 9, 0), (16*a +
11, 17*a + 10, 0), (18*a + 1, 0, 0), (18*a + 1, 12*a + 16, 0), (18*a +
1, 7*a + 3, 0), (a + 10, 0, 0), (a + 10, 16*a + 15, 0), (a + 10, 3*a +
4, 0), (3*a, 0, 0), (3*a, 6*a + 8, 0), (3*a, 13*a + 11, 0), (5*a + 9, 0,
0), (5*a + 9, 4*a + 18, 0), (5*a + 9, 15*a + 1, 0), (7*a + 18, 0, 0),
(7*a + 18, 14*a + 6, 0), (7*a + 18, 5*a + 13, 0), (9*a + 8, 0, 0), (9*a
+ 8, 14*a + 6, 0), (9*a + 8, 5*a + 13, 0), (11*a + 17, 0, 0), (11*a +
17, 4*a + 18, 0), (11*a + 17, 15*a + 1, 0), (13*a + 7, 0, 0), (13*a + 7,
6*a + 8, 0), (13*a + 7, 13*a + 11, 0), (15*a + 16, 0, 0), (15*a + 16,
16*a + 15, 0), (15*a + 16, 3*a + 4, 0), (17*a + 6, 0, 0), (17*a + 6,
12*a + 16, 0), (17*a + 6, 7*a + 3, 0), (15, 0, 0), (15, 2*a + 9, 0),
(15, 17*a + 10, 0), (2*a + 5, 0, 0), (2*a + 5, 8*a + 17, 0), (2*a + 5,
11*a + 2, 0), (4*a + 14, 0, 0), (4*a + 14, 18*a + 5, 0), (4*a + 14, a +
14, 0), (6*a + 4, 0, 0), (6*a + 4, 10*a + 7, 0), (6*a + 4, 9*a + 12, 0),
(6*a + 5, 5*a + 13, 2*a + 9), (12*a + 13, 6*a + 8, 2*a + 9), (12*a + 13,
16*a + 15, 2*a + 9), (18*a + 2, 7*a + 3, 2*a + 9), (5*a + 10, 12*a + 16,
2*a + 9), (5*a + 10, a + 14, 2*a + 9), (5*a + 10, 15*a + 1, 2*a + 9),
(7*a, 4*a + 18, 2*a + 9), (9*a + 9, 17*a + 10, 2*a + 9), (11*a + 18,
18*a + 5, 2*a + 9), (13*a + 8, 0, 2*a + 9), (13*a + 8, 6*a + 8, 2*a +
9), (13*a + 8, 9*a + 12, 2*a + 9), (15*a + 17, 3*a + 4, 2*a + 9), (17*a
+ 7, 10*a + 7, 2*a + 9), (17*a + 7, 3*a + 4, 2*a + 9), (4*a + 15, 3*a +
4, 2*a + 9), (6*a + 5, 0, 0), (8*a + 14, 0, 0), (8*a + 14, 10*a + 7, 0),
(8*a + 14, 9*a + 12, 0), (10*a + 4, 0, 0), (10*a + 4, 18*a + 5, 0),
(10*a + 4, a + 14, 0), (12*a + 13, 0, 0), (12*a + 13, 8*a + 17, 0),
(12*a + 13, 11*a + 2, 0), (14*a + 3, 0, 0), (14*a + 3, 2*a + 9, 0),
(14*a + 3, 17*a + 10, 0), (16*a + 12, 0, 0), (16*a + 12, 12*a + 16, 0),
(16*a + 12, 7*a + 3, 0), (18*a + 2, 0, 0), (18*a + 2, 16*a + 15, 0),
(18*a + 2, 3*a + 4, 0), (a + 11, 0, 0), (a + 11, 6*a + 8, 0), (a + 11,
13*a + 11, 0), (3*a + 1, 0, 0), (3*a + 1, 4*a + 18, 0), (3*a + 1, 15*a +
1, 0), (5*a + 10, 0, 0), (5*a + 10, 14*a + 6, 0), (5*a + 10, 5*a + 13,
0), (7*a, 0, 0), (7*a, 14*a + 6, 0), (7*a, 5*a + 13, 0), (9*a + 9, 0,
0), (9*a + 9, 4*a + 18, 0), (9*a + 9, 15*a + 1, 0), (11*a + 18, 0, 0),
(11*a + 18, 6*a + 8, 0), (11*a + 18, 13*a + 11, 0), (13*a + 8, 0, 0),
(13*a + 8, 16*a + 15, 0), (13*a + 8, 3*a + 4, 0), (15*a + 17, 0, 0),
(15*a + 17, 12*a + 16, 0), (15*a + 17, 7*a + 3, 0), (17*a + 7, 0, 0),
(17*a + 7, 2*a + 9, 0), (17*a + 7, 17*a + 10, 0), (16, 0, 0), (16, 8*a +
17, 0), (16, 11*a + 2, 0), (2*a + 6, 0, 0), (2*a + 6, 18*a + 5, 0), (2*a
+ 6, a + 14, 0), (4*a + 15, 0, 0), (4*a + 15, 10*a + 7, 0), (4*a + 15,
9*a + 12, 0), (4*a + 16, 5*a + 13, 2*a + 9), (10*a + 5, 6*a + 8, 2*a +
9), (10*a + 5, 16*a + 15, 2*a + 9), (16*a + 13, 7*a + 3, 2*a + 9), (3*a
+ 2, 12*a + 16, 2*a + 9), (3*a + 2, a + 14, 2*a + 9), (3*a + 2, 15*a +
1, 2*a + 9), (5*a + 11, 4*a + 18, 2*a + 9), (7*a + 1, 17*a + 10, 2*a +
9), (9*a + 10, 18*a + 5, 2*a + 9), (11*a, 0, 2*a + 9), (11*a, 6*a + 8,
2*a + 9), (11*a, 9*a + 12, 2*a + 9), (13*a + 9, 3*a + 4, 2*a + 9), (15*a
+ 18, 10*a + 7, 2*a + 9), (15*a + 18, 3*a + 4, 2*a + 9), (2*a + 7, 3*a +
4, 2*a + 9), (4*a + 16, 0, 0), (6*a + 6, 0, 0), (6*a + 6, 10*a + 7, 0),
(6*a + 6, 9*a + 12, 0), (8*a + 15, 0, 0), (8*a + 15, 18*a + 5, 0), (8*a
+ 15, a + 14, 0), (10*a + 5, 0, 0), (10*a + 5, 8*a + 17, 0), (10*a + 5,
11*a + 2, 0), (12*a + 14, 0, 0), (12*a + 14, 2*a + 9, 0), (12*a + 14,
17*a + 10, 0), (14*a + 4, 0, 0), (14*a + 4, 12*a + 16, 0), (14*a + 4,
7*a + 3, 0), (16*a + 13, 0, 0), (16*a + 13, 16*a + 15, 0), (16*a + 13,
3*a + 4, 0), (18*a + 3, 0, 0), (18*a + 3, 6*a + 8, 0), (18*a + 3, 13*a +
11, 0), (a + 12, 0, 0), (a + 12, 4*a + 18, 0), (a + 12, 15*a + 1, 0),
(3*a + 2, 0, 0), (3*a + 2, 14*a + 6, 0), (3*a + 2, 5*a + 13, 0), (5*a +
11, 0, 0), (5*a + 11, 14*a + 6, 0), (5*a + 11, 5*a + 13, 0), (7*a + 1,
0, 0), (7*a + 1, 4*a + 18, 0), (7*a + 1, 15*a + 1, 0), (9*a + 10, 0, 0),
(9*a + 10, 6*a + 8, 0), (9*a + 10, 13*a + 11, 0), (11*a, 0, 0), (11*a,
16*a + 15, 0), (11*a, 3*a + 4, 0), (13*a + 9, 0, 0), (13*a + 9, 12*a +
16, 0), (13*a + 9, 7*a + 3, 0), (15*a + 18, 0, 0), (15*a + 18, 2*a + 9,
0), (15*a + 18, 17*a + 10, 0), (17*a + 8, 0, 0), (17*a + 8, 8*a + 17,
0), (17*a + 8, 11*a + 2, 0), (17, 0, 0), (17, 18*a + 5, 0), (17, a + 14,
0), (2*a + 7, 0, 0), (2*a + 7, 10*a + 7, 0), (2*a + 7, 9*a + 12, 0),
(2*a + 8, 5*a + 13, 2*a + 9), (8*a + 16, 6*a + 8, 2*a + 9), (8*a + 16,
16*a + 15, 2*a + 9), (14*a + 5, 7*a + 3, 2*a + 9), (a + 13, 12*a + 16,
2*a + 9), (a + 13, a + 14, 2*a + 9), (a + 13, 15*a + 1, 2*a + 9), (3*a +
3, 4*a + 18, 2*a + 9), (5*a + 12, 17*a + 10, 2*a + 9), (7*a + 2, 18*a +
5, 2*a + 9), (9*a + 11, 0, 2*a + 9), (9*a + 11, 6*a + 8, 2*a + 9), (9*a
+ 11, 9*a + 12, 2*a + 9), (11*a + 1, 3*a + 4, 2*a + 9), (13*a + 10, 10*a
+ 7, 2*a + 9), (13*a + 10, 3*a + 4, 2*a + 9), (18, 3*a + 4, 2*a + 9),
(2*a + 8, 0, 0), (4*a + 17, 0, 0), (4*a + 17, 10*a + 7, 0), (4*a + 17,
9*a + 12, 0), (6*a + 7, 0, 0), (6*a + 7, 18*a + 5, 0), (6*a + 7, a + 14,
0), (8*a + 16, 0, 0), (8*a + 16, 8*a + 17, 0), (8*a + 16, 11*a + 2, 0),
(10*a + 6, 0, 0), (10*a + 6, 2*a + 9, 0), (10*a + 6, 17*a + 10, 0),
(12*a + 15, 0, 0), (12*a + 15, 12*a + 16, 0), (12*a + 15, 7*a + 3, 0),
(14*a + 5, 0, 0), (14*a + 5, 16*a + 15, 0), (14*a + 5, 3*a + 4, 0),
(16*a + 14, 0, 0), (16*a + 14, 6*a + 8, 0), (16*a + 14, 13*a + 11, 0),
(18*a + 4, 0, 0), (18*a + 4, 4*a + 18, 0), (18*a + 4, 15*a + 1, 0), (a +
13, 0, 0), (a + 13, 14*a + 6, 0), (a + 13, 5*a + 13, 0), (3*a + 3, 0,
0), (3*a + 3, 14*a + 6, 0), (3*a + 3, 5*a + 13, 0), (5*a + 12, 0, 0),
(5*a + 12, 4*a + 18, 0), (5*a + 12, 15*a + 1, 0), (7*a + 2, 0, 0), (7*a
+ 2, 6*a + 8, 0), (7*a + 2, 13*a + 11, 0), (9*a + 11, 0, 0), (9*a + 11,
16*a + 15, 0), (9*a + 11, 3*a + 4, 0), (11*a + 1, 0, 0), (11*a + 1, 12*a
+ 16, 0), (11*a + 1, 7*a + 3, 0), (13*a + 10, 0, 0), (13*a + 10, 2*a +
9, 0), (13*a + 10, 17*a + 10, 0), (15*a, 0, 0), (15*a, 8*a + 17, 0),
(15*a, 11*a + 2, 0), (17*a + 9, 0, 0), (17*a + 9, 18*a + 5, 0), (17*a +
9, a + 14, 0), (18, 0, 0), (18, 10*a + 7, 0), (18, 9*a + 12, 0)]

WARNING: Output truncated!  
full_output.txt



[(0, 4*a + 9, 1), (6*a + 8, 5*a + 4, 1), (6*a + 8, 15*a + 11, 1), (12*a + 16, 6*a + 18, 1), (18*a + 5, 11*a + 12, 1), (18*a + 5, 10, 1), (18*a + 5, 14*a + 16, 1), (a + 14, 3*a + 14, 1), (3*a + 4, 16*a + 6, 1), (5*a + 13, 17*a + 1, 1), (7*a + 3, 18*a + 15, 1), (7*a + 3, 5*a + 4, 1), (7*a + 3, 8*a + 8, 1), (9*a + 12, 2*a, 1), (11*a + 2, 9*a + 3, 1), (11*a + 2, 2*a, 1), (17*a + 10, 2*a, 1), (0, 18*a + 15, 17*a + 11), (2*a + 9, 18*a + 15, 17*a + 11), (2*a + 9, 9*a + 3, 17*a + 11), (2*a + 9, 8*a + 8, 17*a + 11), (4*a + 18, 18*a + 15, 17*a + 11), (4*a + 18, 17*a + 1, 17*a + 11), (4*a + 18, 10, 17*a + 11), (6*a + 8, 18*a + 15, 17*a + 11), (6*a + 8, 7*a + 13, 17*a + 11), (6*a + 8, 10*a + 17, 17*a + 11), (8*a + 17, 18*a + 15, 17*a + 11), (8*a + 17, a + 5, 17*a + 11), (8*a + 17, 16*a + 6, 17*a + 11), (10*a + 7, 18*a + 15, 17*a + 11), (10*a + 7, 11*a + 12, 17*a + 11), (10*a + 7, 6*a + 18, 17*a + 11), (12*a + 16, 18*a + 15, 17*a + 11), (12*a + 16, 15*a + 11, 17*a + 11), (12*a + 16, 2*a, 17*a + 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(13*a + 5, 4*a + 18, 2*a + 9), (15*a + 14, 17*a + 10, 2*a + 9), (17*a + 4, 18*a + 5, 2*a + 9), (13, 0, 2*a + 9), (13, 6*a + 8, 2*a + 9), (13, 9*a + 12, 2*a + 9), (2*a + 3, 3*a + 4, 2*a + 9), (4*a + 12, 10*a + 7, 2*a + 9), (4*a + 12, 3*a + 4, 2*a + 9), (10*a + 1, 3*a + 4, 2*a + 9), (12*a + 10, 0, 0), (14*a, 0, 0), (14*a, 10*a + 7, 0), (14*a, 9*a + 12, 0), (16*a + 9, 0, 0), (16*a + 9, 18*a + 5, 0), (16*a + 9, a + 14, 0), (18*a + 18, 0, 0), (18*a + 18, 8*a + 17, 0), (18*a + 18, 11*a + 2, 0), (a + 8, 0, 0), (a + 8, 2*a + 9, 0), (a + 8, 17*a + 10, 0), (3*a + 17, 0, 0), (3*a + 17, 12*a + 16, 0), (3*a + 17, 7*a + 3, 0), (5*a + 7, 0, 0), (5*a + 7, 16*a + 15, 0), (5*a + 7, 3*a + 4, 0), (7*a + 16, 0, 0), (7*a + 16, 6*a + 8, 0), (7*a + 16, 13*a + 11, 0), (9*a + 6, 0, 0), (9*a + 6, 4*a + 18, 0), (9*a + 6, 15*a + 1, 0), (11*a + 15, 0, 0), (11*a + 15, 14*a + 6, 0), (11*a + 15, 5*a + 13, 0), (13*a + 5, 0, 0), (13*a + 5, 14*a + 6, 0), (13*a + 5, 5*a + 13, 0), (15*a + 14, 0, 0), (15*a + 14, 4*a + 18, 0), (15*a + 14, 15*a + 1, 0), (17*a + 4, 0, 0), (17*a + 4, 6*a + 8, 0), (17*a + 4, 13*a + 11, 0), (13, 0, 0), (13, 16*a + 15, 0), (13, 3*a + 4, 0), (2*a + 3, 0, 0), (2*a + 3, 12*a + 16, 0), (2*a + 3, 7*a + 3, 0), (4*a + 12, 0, 0), (4*a + 12, 2*a + 9, 0), (4*a + 12, 17*a + 10, 0), (6*a + 2, 0, 0), (6*a + 2, 8*a + 17, 0), (6*a + 2, 11*a + 2, 0), (8*a + 11, 0, 0), (8*a + 11, 18*a + 5, 0), (8*a + 11, a + 14, 0), (10*a + 1, 0, 0), (10*a + 1, 10*a + 7, 0), (10*a + 1, 9*a + 12, 0), (10*a + 2, 5*a + 13, 2*a + 9), (16*a + 10, 6*a + 8, 2*a + 9), (16*a + 10, 16*a + 15, 2*a + 9), (3*a + 18, 7*a + 3, 2*a + 9), (9*a + 7, 12*a + 16, 2*a + 9), (9*a + 7, a + 14, 2*a + 9), (9*a + 7, 15*a + 1, 2*a + 9), (11*a + 16, 4*a + 18, 2*a + 9), (13*a + 6, 17*a + 10, 2*a + 9), (15*a + 15, 18*a + 5, 2*a + 9), (17*a + 5, 0, 2*a + 9), (17*a + 5, 6*a + 8, 2*a + 9), (17*a + 5, 9*a + 12, 2*a + 9), (14, 3*a + 4, 2*a + 9), (2*a + 4, 10*a + 7, 2*a + 9), (2*a + 4, 3*a + 4, 2*a + 9), (8*a + 12, 3*a + 4, 2*a + 9), (10*a + 2, 0, 0), (12*a + 11, 0, 0), (12*a + 11, 10*a + 7, 0), (12*a + 11, 9*a + 12, 0), (14*a + 1, 0, 0), (14*a + 1, 18*a + 5, 0), (14*a + 1, a + 14, 0), (16*a + 10, 0, 0), (16*a + 10, 8*a + 17, 0), (16*a + 10, 11*a + 2, 0), (18*a, 0, 0), (18*a, 2*a + 9, 0), (18*a, 17*a + 10, 0), (a + 9, 0, 0), (a + 9, 12*a + 16, 0), (a + 9, 7*a + 3, 0), (3*a + 18, 0, 0), (3*a + 18, 16*a + 15, 0), (3*a + 18, 3*a + 4, 0), (5*a + 8, 0, 0), (5*a + 8, 6*a + 8, 0), (5*a + 8, 13*a + 11, 0), (7*a + 17, 0, 0), (7*a + 17, 4*a + 18, 0), (7*a + 17, 15*a + 1, 0), (9*a + 7, 0, 0), (9*a + 7, 14*a + 6, 0), (9*a + 7, 5*a + 13, 0), (11*a + 16, 0, 0), (11*a + 16, 14*a + 6, 0), (11*a + 16, 5*a + 13, 0), (13*a + 6, 0, 0), (13*a + 6, 4*a + 18, 0), (13*a + 6, 15*a + 1, 0), (15*a + 15, 0, 0), (15*a + 15, 6*a + 8, 0), (15*a + 15, 13*a + 11, 0), (17*a + 5, 0, 0), (17*a + 5, 16*a + 15, 0), (17*a + 5, 3*a + 4, 0), (14, 0, 0), (14, 12*a + 16, 0), (14, 7*a + 3, 0), (2*a + 4, 0, 0), (2*a + 4, 2*a + 9, 0), (2*a + 4, 17*a + 10, 0), (4*a + 13, 0, 0), (4*a + 13, 8*a + 17, 0), (4*a + 13, 11*a + 2, 0), (6*a + 3, 0, 0), (6*a + 3, 18*a + 5, 0), (6*a + 3, a + 14, 0), (8*a + 12, 0, 0), (8*a + 12, 10*a + 7, 0), (8*a + 12, 9*a + 12, 0), (8*a + 13, 5*a + 13, 2*a + 9), (14*a + 2, 6*a + 8, 2*a + 9), (14*a + 2, 16*a + 15, 2*a + 9), (a + 10, 7*a + 3, 2*a + 9), (7*a + 18, 12*a + 16, 2*a + 9), (7*a + 18, a + 14, 2*a + 9), (7*a + 18, 15*a + 1, 2*a + 9), (9*a + 8, 4*a + 18, 2*a + 9), (11*a + 17, 17*a + 10, 2*a + 9), (13*a + 7, 18*a + 5, 2*a + 9), (15*a + 16, 0, 2*a + 9), (15*a + 16, 6*a + 8, 2*a + 9), (15*a + 16, 9*a + 12, 2*a + 9), (17*a + 6, 3*a + 4, 2*a + 9), (15, 10*a + 7, 2*a + 9), (15, 3*a + 4, 2*a + 9), (6*a + 4, 3*a + 4, 2*a + 9), (8*a + 13, 0, 0), (10*a + 3, 0, 0), (10*a + 3, 10*a + 7, 0), (10*a + 3, 9*a + 12, 0), (12*a + 12, 0, 0), (12*a + 12, 18*a + 5, 0), (12*a + 12, a + 14, 0), (14*a + 2, 0, 0), (14*a + 2, 8*a + 17, 0), (14*a + 2, 11*a + 2, 0), (16*a + 11, 0, 0), (16*a + 11, 2*a + 9, 0), (16*a + 11, 17*a + 10, 0), (18*a + 1, 0, 0), (18*a + 1, 12*a + 16, 0), (18*a + 1, 7*a + 3, 0), (a + 10, 0, 0), (a + 10, 16*a + 15, 0), (a + 10, 3*a + 4, 0), (3*a, 0, 0), (3*a, 6*a + 8, 0), (3*a, 13*a + 11, 0), (5*a + 9, 0, 0), (5*a + 9, 4*a + 18, 0), (5*a + 9, 15*a + 1, 0), (7*a + 18, 0, 0), (7*a + 18, 14*a + 6, 0), (7*a + 18, 5*a + 13, 0), (9*a + 8, 0, 0), (9*a + 8, 14*a + 6, 0), (9*a + 8, 5*a + 13, 0), (11*a + 17, 0, 0), (11*a + 17, 4*a + 18, 0), (11*a + 17, 15*a + 1, 0), (13*a + 7, 0, 0), (13*a + 7, 6*a + 8, 0), (13*a + 7, 13*a + 11, 0), (15*a + 16, 0, 0), (15*a + 16, 16*a + 15, 0), (15*a + 16, 3*a + 4, 0), (17*a + 6, 0, 0), (17*a + 6, 12*a + 16, 0), (17*a + 6, 7*a + 3, 0), (15, 0, 0), (15, 2*a + 9, 0), (15, 17*a + 10, 0), (2*a + 5, 0, 0), (2*a + 5, 8*a + 17, 0), (2*a + 5, 11*a + 2, 0), (4*a + 14, 0, 0), (4*a + 14, 18*a + 5, 0), (4*a + 14, a + 14, 0), (6*a + 4, 0, 0), (6*a + 4, 10*a + 7, 0), (6*a + 4, 9*a + 12, 0), (6*a + 5, 5*a + 13, 2*a + 9), (12*a + 13, 6*a + 8, 2*a + 9), (12*a + 13, 16*a + 15, 2*a + 9), (18*a + 2, 7*a + 3, 2*a + 9), (5*a + 10, 12*a + 16, 2*a + 9), (5*a + 10, a + 14, 2*a + 9), (5*a + 10, 15*a + 1, 2*a + 9), (7*a, 4*a + 18, 2*a + 9), (9*a + 9, 17*a + 10, 2*a + 9), (11*a + 18, 18*a + 5, 2*a + 9), (13*a + 8, 0, 2*a + 9), (13*a + 8, 6*a + 8, 2*a + 9), (13*a + 8, 9*a + 12, 2*a + 9), (15*a + 17, 3*a + 4, 2*a + 9), (17*a + 7, 10*a + 7, 2*a + 9), (17*a + 7, 3*a + 4, 2*a + 9), (4*a + 15, 3*a + 4, 2*a + 9), (6*a + 5, 0, 0), (8*a + 14, 0, 0), (8*a + 14, 10*a + 7, 0), (8*a + 14, 9*a + 12, 0), (10*a + 4, 0, 0), (10*a + 4, 18*a + 5, 0), (10*a + 4, a + 14, 0), (12*a + 13, 0, 0), (12*a + 13, 8*a + 17, 0), (12*a + 13, 11*a + 2, 0), (14*a + 3, 0, 0), (14*a + 3, 2*a + 9, 0), (14*a + 3, 17*a + 10, 0), (16*a + 12, 0, 0), (16*a + 12, 12*a + 16, 0), (16*a + 12, 7*a + 3, 0), (18*a + 2, 0, 0), (18*a + 2, 16*a + 15, 0), (18*a + 2, 3*a + 4, 0), (a + 11, 0, 0), (a + 11, 6*a + 8, 0), (a + 11, 13*a + 11, 0), (3*a + 1, 0, 0), (3*a + 1, 4*a + 18, 0), (3*a + 1, 15*a + 1, 0), (5*a + 10, 0, 0), (5*a + 10, 14*a + 6, 0), (5*a + 10, 5*a + 13, 0), (7*a, 0, 0), (7*a, 14*a + 6, 0), (7*a, 5*a + 13, 0), (9*a + 9, 0, 0), (9*a + 9, 4*a + 18, 0), (9*a + 9, 15*a + 1, 0), (11*a + 18, 0, 0), (11*a + 18, 6*a + 8, 0), (11*a + 18, 13*a + 11, 0), (13*a + 8, 0, 0), (13*a + 8, 16*a + 15, 0), (13*a + 8, 3*a + 4, 0), (15*a + 17, 0, 0), (15*a + 17, 12*a + 16, 0), (15*a + 17, 7*a + 3, 0), (17*a + 7, 0, 0), (17*a + 7, 2*a + 9, 0), (17*a + 7, 17*a + 10, 0), (16, 0, 0), (16, 8*a + 17, 0), (16, 11*a + 2, 0), (2*a + 6, 0, 0), (2*a + 6, 18*a + 5, 0), (2*a + 6, a + 14, 0), (4*a + 15, 0, 0), (4*a + 15, 10*a + 7, 0), (4*a + 15, 9*a + 12, 0), (4*a + 16, 5*a + 13, 2*a + 9), (10*a + 5, 6*a + 8, 2*a + 9), (10*a + 5, 16*a + 15, 2*a + 9), (16*a + 13, 7*a + 3, 2*a + 9), (3*a + 2, 12*a + 16, 2*a + 9), (3*a + 2, a + 14, 2*a + 9), (3*a + 2, 15*a + 1, 2*a + 9), (5*a + 11, 4*a + 18, 2*a + 9), (7*a + 1, 17*a + 10, 2*a + 9), (9*a + 10, 18*a + 5, 2*a + 9), (11*a, 0, 2*a + 9), (11*a, 6*a + 8, 2*a + 9), (11*a, 9*a + 12, 2*a + 9), (13*a + 9, 3*a + 4, 2*a + 9), (15*a + 18, 10*a + 7, 2*a + 9), (15*a + 18, 3*a + 4, 2*a + 9), (2*a + 7, 3*a + 4, 2*a + 9), (4*a + 16, 0, 0), (6*a + 6, 0, 0), (6*a + 6, 10*a + 7, 0), (6*a + 6, 9*a + 12, 0), (8*a + 15, 0, 0), (8*a + 15, 18*a + 5, 0), (8*a + 15, a + 14, 0), (10*a + 5, 0, 0), (10*a + 5, 8*a + 17, 0), (10*a + 5, 11*a + 2, 0), (12*a + 14, 0, 0), (12*a + 14, 2*a + 9, 0), (12*a + 14, 17*a + 10, 0), (14*a + 4, 0, 0), (14*a + 4, 12*a + 16, 0), (14*a + 4, 7*a + 3, 0), (16*a + 13, 0, 0), (16*a + 13, 16*a + 15, 0), (16*a + 13, 3*a + 4, 0), (18*a + 3, 0, 0), (18*a + 3, 6*a + 8, 0), (18*a + 3, 13*a + 11, 0), (a + 12, 0, 0), (a + 12, 4*a + 18, 0), (a + 12, 15*a + 1, 0), (3*a + 2, 0, 0), (3*a + 2, 14*a + 6, 0), (3*a + 2, 5*a + 13, 0), (5*a + 11, 0, 0), (5*a + 11, 14*a + 6, 0), (5*a + 11, 5*a + 13, 0), (7*a + 1, 0, 0), (7*a + 1, 4*a + 18, 0), (7*a + 1, 15*a + 1, 0), (9*a + 10, 0, 0), (9*a + 10, 6*a + 8, 0), (9*a + 10, 13*a + 11, 0), (11*a, 0, 0), (11*a, 16*a + 15, 0), (11*a, 3*a + 4, 0), (13*a + 9, 0, 0), (13*a + 9, 12*a + 16, 0), (13*a + 9, 7*a + 3, 0), (15*a + 18, 0, 0), (15*a + 18, 2*a + 9, 0), (15*a + 18, 17*a + 10, 0), (17*a + 8, 0, 0), (17*a + 8, 8*a + 17, 0), (17*a + 8, 11*a + 2, 0), (17, 0, 0), (17, 18*a + 5, 0), (17, a + 14, 0), (2*a + 7, 0, 0), (2*a + 7, 10*a + 7, 0), (2*a + 7, 9*a + 12, 0), (2*a + 8, 5*a + 13, 2*a + 9), (8*a + 16, 6*a + 8, 2*a + 9), (8*a + 16, 16*a + 15, 2*a + 9), (14*a + 5, 7*a + 3, 2*a + 9), (a + 13, 12*a + 16, 2*a + 9), (a + 13, a + 14, 2*a + 9), (a + 13, 15*a + 1, 2*a + 9), (3*a + 3, 4*a + 18, 2*a + 9), (5*a + 12, 17*a + 10, 2*a + 9), (7*a + 2, 18*a + 5, 2*a + 9), (9*a + 11, 0, 2*a + 9), (9*a + 11, 6*a + 8, 2*a + 9), (9*a + 11, 9*a + 12, 2*a + 9), (11*a + 1, 3*a + 4, 2*a + 9), (13*a + 10, 10*a + 7, 2*a + 9), (13*a + 10, 3*a + 4, 2*a + 9), (18, 3*a + 4, 2*a + 9), (2*a + 8, 0, 0), (4*a + 17, 0, 0), (4*a + 17, 10*a + 7, 0), (4*a + 17, 9*a + 12, 0), (6*a + 7, 0, 0), (6*a + 7, 18*a + 5, 0), (6*a + 7, a + 14, 0), (8*a + 16, 0, 0), (8*a + 16, 8*a + 17, 0), (8*a + 16, 11*a + 2, 0), (10*a + 6, 0, 0), (10*a + 6, 2*a + 9, 0), (10*a + 6, 17*a + 10, 0), (12*a + 15, 0, 0), (12*a + 15, 12*a + 16, 0), (12*a + 15, 7*a + 3, 0), (14*a + 5, 0, 0), (14*a + 5, 16*a + 15, 0), (14*a + 5, 3*a + 4, 0), (16*a + 14, 0, 0), (16*a + 14, 6*a + 8, 0), (16*a + 14, 13*a + 11, 0), (18*a + 4, 0, 0), (18*a + 4, 4*a + 18, 0), (18*a + 4, 15*a + 1, 0), (a + 13, 0, 0), (a + 13, 14*a + 6, 0), (a + 13, 5*a + 13, 0), (3*a + 3, 0, 0), (3*a + 3, 14*a + 6, 0), (3*a + 3, 5*a + 13, 0), (5*a + 12, 0, 0), (5*a + 12, 4*a + 18, 0), (5*a + 12, 15*a + 1, 0), (7*a + 2, 0, 0), (7*a + 2, 6*a + 8, 0), (7*a + 2, 13*a + 11, 0), (9*a + 11, 0, 0), (9*a + 11, 16*a + 15, 0), (9*a + 11, 3*a + 4, 0), (11*a + 1, 0, 0), (11*a + 1, 12*a + 16, 0), (11*a + 1, 7*a + 3, 0), (13*a + 10, 0, 0), (13*a + 10, 2*a + 9, 0), (13*a + 10, 17*a + 10, 0), (15*a, 0, 0), (15*a, 8*a + 17, 0), (15*a, 11*a + 2, 0), (17*a + 9, 0, 0), (17*a + 9, 18*a + 5, 0), (17*a + 9, a + 14, 0), (18, 0, 0), (18, 10*a + 7, 0), (18, 9*a + 12, 0)]

full_output.txt

len(Llist)*len(Rlist)

Rlist

[(0, 4, 1), (1, 0, 1), (1, 8, 1), (1, 10, 1), (2, 4, 1), (2, 6, 1), (2,
7, 1), (3, 9, 1), (5, 1, 1), (6, 7, 1), (7, 8, 1), (8, 4, 1), (10, 2,
1), (10, 8, 1), (0, 0, 0), (0, 1, 0), (0, 2, 0), (0, 3, 0), (0, 4, 0),
(0, 5, 0), (0, 6, 0), (0, 7, 0), (0, 8, 0), (0, 9, 0), (0, 10, 0), (1,
0, 0), (2, 0, 0), (3, 0, 0), (4, 0, 0), (5, 0, 0), (6, 0, 0), (7, 0, 0),
(8, 0, 0), (9, 0, 0), (10, 0, 0)]

[(0, 4, 1), (1, 0, 1), (1, 8, 1), (1, 10, 1), (2, 4, 1), (2, 6, 1), (2, 7, 1), (3, 9, 1), (5, 1, 1), (6, 7, 1), (7, 8, 1), (8, 4, 1), (10, 2, 1), (10, 8, 1), (0, 0, 0), (0, 1, 0), (0, 2, 0), (0, 3, 0), (0, 4, 0), (0, 5, 0), (0, 6, 0), (0, 7, 0), (0, 8, 0), (0, 9, 0), (0, 10, 0), (1, 0, 0), (2, 0, 0), (3, 0, 0), (4, 0, 0), (5, 0, 0), (6, 0, 0), (7, 0, 0), (8, 0, 0), (9, 0, 0), (10, 0, 0)]

R(768*disc)

from psage.number_fields.sqrt5.misc import F R = F.residue_field(primes_of_bounded_norm(12)[-2].sage_ideal()) L = F.residue_field(primes_of_bounded_norm(12)[-1].sage_ideal())

primes_of_bounded_norm(20)[-2].r

mod(10,19)^-1

R(disc)

disc = -63 + 9*a c4 = 32 + 80*a c6 = -816 - 776*a F = lambda x,y,z: 12*x^3*z + 108*x^2*y^2 + 3*c4*x^2*z^2 + 72*c4*x*y^2*z - 108*c4*y^4 - 12*c6*x*y*z^2 +84*c6*y^3*z + c4^2*x*z^3 - 15*c4^2*y^2*z^2 + c4*c6*y*z^3 + 768*disc*z^4 G = lambda x,y,z: 3*x^4 + c4*x^3*z - 18*c4*x^2*y^2 - 3*c6*x^2*y*z + 24*c6*x*y^3 + 3*c4^2*x*y^2*z - 9*c4^2*y^4 - c4*c6*y^3*z + 168*disc*x*z^3 + 1728*disc*y^2*z^2 + 5*c4*disc*z^4

F(-12*a - 12, -3, 0)

-279936*a

-279936*a

G(-10*a - 10, -a - 5, -8*a + 5)

199226592*a

199226592*a

len(199226592.bits())

Fisher_Method_c++

286 days ago by andrew.ohana