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I was able to cut the time on my computer in half

The first OUTPUT was

(x, y, p, a, A, b, B, c, C, d, D, f, z, Z, w, W, N, v, V, u, U, t, T)
7.9771857261657715
[
[x == -6, y == 10, p == -41, a == -11, A == 45, b == -5, B == 57, c == 3, C == 59, d == -1, D == 60, f == 1, z == -9, Z == 35, w == 5, W == 7, N == 123, v == 3, V == 2, u == -1, U == 1, t == 1, T == 0],
[x == -6, y == -9, p == -3, a == -11, A == 45, b == -5, B == 57, c == 3, C == 59, d == -1, D == 60, f == 1, z == -9, Z == 35, w == 5, W == 7, N == 123, v == 3, V == 2, u == -1, U == 1, t == 1, T == 0],
[x == 5, y == 10, p == 3, a == 11, A == 45, b == -5, B == 57, c == 3, C == 59, d == -1, D == 60, f == 1, z == -9, Z == 35, w == 5, W == 7, N == 123, v == 3, V == 2, u == -1, U == 1, t == 1, T == 0],
[x == 5, y == -9, p == 41, a == 11, A == 45, b == -5, B == 57, c == 3, C == 59, d == -1, D == 60, f == 1, z == -9, Z == 35, w == 5, W == 7, N == 123, v == 3, V == 2, u == -1, U == 1, t == 1, T == 0],
[x == -16, y == 31, p == -123, a == -31, A == 360, b == -15, B == 452, c == -7, C == 474, d == -3, D == 479, f == -1, z == 31, Z == 345, w == -15, W == 92, N == 123, v == -7, V == 22, u == -3, U == 5, t == -1, T == 1],
[x == -16, y == -30, p == -1, a == -31, A == 360, b == -15, B == 452, c == -7, C == 474, d == -3, D == 479, f == -1, z == 31, Z == 345, w == -15, W == 92, N == 123, v == -7, V == 22, u == -3, U == 5, t == -1, T == 1],
[x == 15, y == 31, p == 1, a == 31, A == 360, b == -15, B == 452, c == -7, C == 474, d == -3, D == 479, f == -1, z == 31, Z == 345, w == -15, W == 92, N == 123, v == -7, V == 22, u == -3, U == 5, t == -1, T == 1],
[x == 15, y == -30, p == 123, a == 31, A == 360, b == -15, B == 452, c == -7, C == 474, d == -3, D == 479, f == -1, z == 31, Z == 345, w == -15, W == 92, N == 123, v == -7, V == 22, u == -3, U == 5, t == -1, T == 1]
]

While now it is OUTPUT

OUTPUT

(N, x, y, p, B, c, C, d, D, e, E, f, F, g, G, h, H, m, L, n)
4.016726016998291
[
[N == 123, x == 15, y == 31, p == 1, B == 452, c == -7, C == 474, d == -3, D == 479, E == 360, e == 31, f == -15, F == 452, g == -7, G == 474, h == -3, H == 480, m == -1, L == 479, n == -1],
[N == 123, x == 15, y == 31, p == 1, B == 452, c == -7, C == 474, d == -3, D == 479, E == 360, e == -30, f == -15, F == 452, g == -7, G == 474, h == -3, H == 480, m == -1, L == 479, n == -1],
[N == 123, x == 5, y == -9, p == 41, B == 57, c == 3, C == 59, d == -1, D == 60, E == 50, e == 10, f == 5, F == 57, g == 3, G == 59, h == -1, H == 60, m == 1, L == 60, n == 1],
[N == 123, x == 5, y == -9, p == 41, B == 57, c == 3, C == 59, d == -1, D == 60, E == 50, e == -9, f == 5, F == 57, g == 3, G == 59, h == -1, H == 60, m == 1, L == 60, n == 1]
]

this is the modified system

import time
Start_Time = time.time()
var('N x y p B c C d D e E f F g G h H m L n')

eq0 = N - 123 == 0


eq1 = (2*(3*(3/2*x*(x+1))+1)-3*(-x)+1)/24+(3/4*(H)) - B == 0
eq2 = B+c*(c-1)/2 - (H) == 0
eq3 = (2*(3*(B)+1)-3*(c)+1)/24+(3/4*(H)) - C == 0
eq4 = C+d*(d-1)/2 - (H) == 0
eq5 = (2*(3*(C)+1)-3*(d)+1)/24+(3/4*(H)) - D == 0
eq19 = D+n*(n-1)/2 - (H) == 0
eq20 = (2*(3*(D)+1)-3*(n)+1)/24+(3/4*(H)) - H == 0



eq6 = (N-3)/8+e*(e-1)/2 - (H) == 0
eq7 = (2*(3*((N-3)/8)+1)-3*(y)+1)/24+(3/4*(H)) - E == 0
eq8 = E+f*(f-1)/2 - (H) == 0
eq9 = (2*(3*(E)+1)-3*(f)+1)/24+(3/4*(H)) - F == 0
eq10 = F+g*(g-1)/2 - (H) == 0
eq11 = (2*(3*(F)+1)-3*(g)+1)/24+(3/4*(H)) - G == 0
eq12 = G+h*(h-1)/2 - (H) == 0
eq13 = (2*(3*(G)+1)-3*(h)+1)/24+(3/4*(H)) - L == 0
eq14 = L+m*(m-1)/2 - (H) == 0
eq17 = (2*(3*(L)+1)-3*(m)+1)/24+(3/4*(H)) - H == 0
eq18 = H - ((N-3)/8+y*(y-1)/2) == 0

eq15 = (N-3)/8+y*(y-1)/2 - 2*x*(x+1) == 0
eq16 = 4*x+1-2*(y-1)-p == 0


solutions = solve([eq0,eq1,eq2,eq3,eq4,eq5,eq6,eq7,eq8,eq9,eq10,eq11,eq12,eq13,eq14,eq15,eq16,eq17,eq18,eq19,eq20],N,x,y,p,B,c,C,d,D,E,e,f,F,g,G,h,H,m,L,n)
sol = solutions  
Execution_Time = time.time() - Start_Time
print (Execution_Time)
print(sol)