This Mathematica notebook contains the formulas to accompany "Rank frequencies for quadratic twists of elliptic curves", by K. Rubin and A. Silverberg.

In each example, the points ((Subscript[x, i]), (Subscript[y, i])) are independent points of infinite order on the elliptic curve g(u) (y^2) =  f(x).

Rank 2 examples

Corollary 4.2

\(f = x\^3 + a\ x\^2 + b\ x; 

    g = \(-a\)\ b\ \((b\^2 + u\^2)\)\ 
        \((u\^4 + \((2\ b\^2 - a\^2\ b)\)\ u\^2 + b\^4)\); 
    
{x\_1, y\_1} = {\(-\(\(b\^2 + u\^2\)\/\(a\ b\)\)\), 1\/\(a\^2\ b\^2\)}; 
    
{x\_2, y\_2} = {\(-\(\(b\ \((b\^2 + u\^2)\)\)\/\(a\ u\^2\)\)\), 
        b\/\(a\^2\ u\^3\)}; \)

Simplify[g ((Subscript[y, 1])^2) - (f /. x -> (Subscript[x, 1]))]

Simplify[g ((Subscript[y, 2])^2) - (f /. x -> (Subscript[x, 2]))]

Corollary 4.4

\(f = x\^3 + b\^2\/\(4  c\)\ x\^2 + b\ x + c; 

    g = \(-b\)\ c\ 
        \((2\ u\^6 + \((18\ c\^2 - b\^3)\)\ u\^4 + 
            \((54\ c\^4 + 2\ b\^3\ c\^2)\)\ u\^2 + 54\ c\^6 - b\^3\ c\^4)\); 
    
{x\_1, y\_1} = {\(-\(\(3\ c\^2 + u\^2\)\/\(2\ b\ c\)\)\), 
        1\/\(4\ b\^2\ c\^2\)}; 
    
{x\_2, y\_2} = {
        \((b\^3\ c\^6 - 54\ c\^8 + 
              \((\(-3\)\ b\^3\ c\^4 - 54\ c\^6)\)\ u\^2 + 
              \((3\ b\^3\ c\^2 - 18\ c\^4)\)\ u\^4 + 
              \((\(-b\^3\) - 2\ c\^2)\)\ u\^6)\)/
          \((4\ b\ c\ u\^2\ \((3\ c\^2 + u\^2)\)\^2)\), 
        \((b\^3\ c\^6 - 54\ c\^8 + \((b\^3\ c\^4 - 54\ c\^6)\)\ u\^2 + 
              \((\(-5\)\ b\^3\ c\^2 - 18\ c\^4)\)\ u\^4 + 
              \((3\ b\^3 - 2\ c\^2)\)\ u\^6)\)/
          \((8\ b\^2\ c\ u\^3\ \((3\ c\^2 + u\^2)\)\^3)\)}; \)

Simplify[g ((Subscript[y, 1])^2) - (f /. x -> (Subscript[x, 1]))]

Simplify[g ((Subscript[y, 2])^2) - (f /. x -> (Subscript[x, 2]))]

Theorem 4.6

\(f = x\^3 + a\ x + b; 

    g = \(-a\)\ b 
        \((\(b\^2\) \((u\^4 + u\^2 + 1)\)\^3 + 
            \(a\^3\) \(u\^4\) \((u\^2 + 1)\)\^2)\) \((u\^2 + 1)\); 
    
{x\_1, y\_1} = {
        \(-\(\(b\ \((\(-1\) + u\^6)\)\)\/\(a\ \((\(-1\) + u\^4)\)\)\)\), 
        1\/\(a\^2\ \((1 + u\^2)\)\^2\)}; 
    
{x\_2, y\_2} = {
        \(-\(\(b\ \((\(-1\) + u\^6)\)\)\/\(a\ u\^2\ 
                \((\(-1\) + u\^4)\)\)\)\), 
        1\/\(a\^2\ u\^3\ \((1 + u\^2)\)\^2\)}; \)

Simplify[g ((Subscript[y, 1])^2) - (f /. x -> (Subscript[x, 1]))]

Simplify[g ((Subscript[y, 2])^2) - (f /. x -> (Subscript[x, 2]))]

Rank 3 examples

Theorem 5.1

\(λ = \(-2\) a\^2; 
f = x \((x - 1)\) \((x - λ)\); \)

\(d = λ \((2  λ - 1)\) u\^2 + 2 - λ; 

    n = \(λ\^2\) \((λ + 1)\) \(\((2  λ - 1)\)\^2\) 
          u\^4 - 4 \( λ\^2\) \((λ - 1)\) 
          \((2  λ - 1)\) u\^3 + 
        2  λ \((λ + 1)\) 
          \((2  λ\^2 - 3  λ + 2)\) u\^2 - 
        4  λ \((λ - 1)\) \((λ - 2)\) u + 
        \(\((λ - 2)\)\^2\) \((λ + 1)\); 

    g = Factor[2  n \((n - 2  d\^2)\) \((n - 2  λ\ d\^2)\)]\)

\({x\_1, y\_1} = {n\/\(2  d\^2\), 1\/\(4\ d\^3\)}; 
    
{x\_2, y\_2} = {
        \(\(λ\^2\) 
            \((d\^2 - 
                4  λ\ u \((u - 1)\) 
                  \((λ \((2  λ - 1)\) u + 2 - λ)\))
              \)\)\/\((
              λ \((2  λ - 1)\) u\^2 - 
                2  λ \((2  λ - 1)\) u + λ - 2)\)\^2, 
        \(a\ λ\)\/\((
              λ \((2  λ - 1)\) u\^2 - 
                2  λ \((2  λ - 1)\) u + λ - 2)\)\^3}; 
    
{x\_3, y\_3} = {
        \(d\^2 + 
            4  λ\ u \((u - 1)\) 
              \((λ \((2  λ - 1)\) u + 2 - λ)
                \)\)\/\(λ 
            \((λ \((2  λ - 1)\) u\^2 - 
                  \((2  λ - 4)\) u + λ - 2)\)\^2\), 
        \(-\(a\/\(\(λ\^2\) 
                \((λ \((2  λ - 1)\) u\^2 - 
                      \((2  λ - 4)\) u + λ - 2)\)\^3\)\)\)}; 
    \)

Factor[g ((Subscript[y, 1])^2) - (f /. x -> (Subscript[x, 1]))]

Factor[g ((Subscript[y, 2])^2) - (f /. x -> (Subscript[x, 2]))]

Factor[g ((Subscript[y, 3])^2) - (f /. x -> (Subscript[x, 3]))]

Theorem 5.2(a)

\(λ = \(1 - a\^2\)\/\(a\^2 + 2\); 

    f = x \((x - 1)\) \((x - λ)\); \)

g = 
      -2 (1 + λ) 
        (4 (u^2) + 8 (u^3) + 4 (u^4) - 4 u λ - 
            16 (u^2) λ - 20 (u^3) λ - 4 (u^4) λ + 
            (λ^2) + 12 u (λ^2) + 26 (u^2) (λ^2) + 
            8 (u^3) (λ^2) + (u^4) (λ^2) - 2 (λ^3) - 
            12 u (λ^3) - 10 (u^2) (λ^3) + 3 (λ^4) + 
            8 u (λ^4) - 2 (u^2) (λ^4) - 2 (λ^5) + 
            (λ^6)) 
        (4 (u^4) - 8 (u^3) λ - 4 (u^4) λ + (λ^2) + 
            4 u (λ^2) + 14 (u^2) (λ^2) + 
            20 (u^3) (λ^2) + (u^4) (λ^2) - 2 (λ^3) - 
            12 u (λ^3) - 26 (u^2) (λ^3) - 
            8 (u^3) (λ^3) + 3 (λ^4) + 12 u (λ^4) + 
            14 (u^2) (λ^4) - 2 (λ^5) - 8 u (λ^5) + 
            (λ^6)) 
        (2 (u^2) + 4 (u^3) + 4 (u^4) - 2 u λ - 
            8 (u^2) λ - 14 (u^3) λ - 4 (u^4) λ + 
            (λ^2) + 8 u (λ^2) + 20 (u^2) (λ^2) + 
            14 (u^3) (λ^2) + (u^4) (λ^2) - 2 (λ^3) - 
            12 u (λ^3) - 18 (u^2) (λ^3) - 
            4 (u^3) (λ^3) + 3 (λ^4) + 10 u (λ^4) + 
            6 (u^2) (λ^4) - 2 (λ^5) - 4 u (λ^5) + 
            (λ^6));

\({x\_1, y\_1} = {
        \((2\ λ\ 
              \((2\ u\^2 + 4\ u\^3 + 4\ u\^4 - 2\ u\ λ - 
                  8\ u\^2\ λ - 14\ u\^3\ λ - 
                  4\ u\^4\ λ + λ\^2 + 8\ u\ λ\^2 + 
                  20\ u\^2\ λ\^2 + 14\ u\^3\ λ\^2 + 
                  u\^4\ λ\^2 - 2\ λ\^3 - 
                  12\ u\ λ\^3 - 18\ u\^2\ λ\^3 - 
                  4\ u\^3\ λ\^3 + 3\ λ\^4 + 
                  10\ u\ λ\^4 + 6\ u\^2\ λ\^4 - 
                  2\ λ\^5 - 4\ u\ λ\^5 + λ\^6)\))\)/
          \((\((1 + λ)\)\ 
              \((2\ u\^2 + λ - u\^2\ λ - λ\^2 + 
                    λ\^3)\)\^2)\), 
        \(-\(\(\((\(-1\) + λ)\)\ 
                λ\)\/\(\((1 + λ)\)\^2\ 
                \((2\ u\^2 + λ - u\^2\ λ - λ\^2 + 
                      λ\^3)\)\^3\)\)\)}; 
    
{x\_2, y\_2} = {
        \((λ\ 
              \((4\ u\^2 + 8\ u\^3 + 4\ u\^4 - 4\ u\ λ - 
                  16\ u\^2\ λ - 20\ u\^3\ λ - 
                  4\ u\^4\ λ + λ\^2 + 12\ u\ λ\^2 + 
                  26\ u\^2\ λ\^2 + 8\ u\^3\ λ\^2 + 
                  u\^4\ λ\^2 - 2\ λ\^3 - 
                  12\ u\ λ\^3 - 10\ u\^2\ λ\^3 + 
                  3\ λ\^4 + 8\ u\ λ\^4 - 
                  2\ u\^2\ λ\^4 - 2\ λ\^5 + λ\^6)\))
            \)/\((\((\(-1\) + 2\ λ)\)\ 
              \((\(-2\)\ u\^2 + λ + 4\ u\ λ + 
                    u\^2\ λ - λ\^2 - 2\ u\ λ\^2 + 
                    λ\^3)\)\^2)\), 
        \(a\ \((\(-1\) + λ)\)\ 
            λ\)\/\(\((\(-1\) + 2\ λ)\)\^2\ 
            \((\(-2\)\ u\^2 + λ + 4\ u\ λ + u\^2\ λ - 
                  λ\^2 - 2\ u\ λ\^2 + λ\^3)\)\^3\)}; 
    
{x\_3, y\_3} = {
        \((λ\ 
              \((4\ u\^4 - 8\ u\^3\ λ - 4\ u\^4\ λ + 
                  λ\^2 + 4\ u\ λ\^2 + 
                  14\ u\^2\ λ\^2 + 20\ u\^3\ λ\^2 + 
                  u\^4\ λ\^2 - 2\ λ\^3 - 
                  12\ u\ λ\^3 - 26\ u\^2\ λ\^3 - 
                  8\ u\^3\ λ\^3 + 3\ λ\^4 + 
                  12\ u\ λ\^4 + 14\ u\^2\ λ\^4 - 
                  2\ λ\^5 - 8\ u\ λ\^5 + λ\^6)\))\)/
          \((\((\(-1\) + 2\ λ)\)\ 
              \((\(-2\)\ u - 2\ u\^2 + λ + 2\ u\ λ + 
                    u\^2\ λ - λ\^2 - 2\ u\ λ\^2 + 
                    λ\^3)\)\^2)\), 
        \(-\(\(a\ \((\(-1\) + λ)\)\ 
                λ\)\/\(\((\(-1\) + 2\ λ)\)\^2\ 
                \((\(-2\)\ u - 2\ u\^2 + λ + 2\ u\ λ + 
                      u\^2\ λ - λ\^2 - 2\ u\ λ\^2 + 
                      λ\^3)\)\^3\)\)\)}; \)

Factor[g ((Subscript[y, 1])^2) - (f /. x -> (Subscript[x, 1]))]

Factor[g ((Subscript[y, 2])^2) - (f /. x -> (Subscript[x, 2]))]

Factor[g ((Subscript[y, 3])^2) - (f /. x -> (Subscript[x, 3]))]

Theorem 5.2(b)

\(λ = \(a \((a - 2)\)\)\/\(a\^2 + 1\); 

    f = x \((x - 1)\) \((x - λ)\); \)

g = 
      (-2 + a) a 
        (1 + 8 a + 34 (a^2) + 88 (a^3) + 139 (a^4) + 124 (a^5) + 
            26 (a^6) - 12 (a^7) + 41 (a^8) - 12 (a^9) + 4 (a^10) - 4 u - 
            20 a u - 44 (a^2) u - 32 (a^3) u + 80 (a^4) u + 
            160 (a^5) u + 140 (a^6) u + 172 (a^7) u + 36 (a^8) u + 
            16 (a^10) u + 6 (u^2) + 12 a (u^2) - 2 (a^2) (u^2) - 
            20 (a^3) (u^2) - 68 (a^4) (u^2) + 20 (a^5) (u^2) + 
            114 (a^6) (u^2) - 4 (a^7) (u^2) + 230 (a^8) (u^2) - 
            56 (a^9) (u^2) + 56 (a^10) (u^2) - 4 (u^3) + 4 a (u^3) - 
            12 (a^3) (u^3) + 32 (a^4) (u^3) - 44 (a^5) (u^3) + 
            56 (a^6) (u^3) - 20 (a^7) (u^3) + 36 (a^8) (u^3) + 
            24 (a^9) (u^3) + 8 (a^10) (u^3) + 16 (a^11) (u^3) + (u^4) - 
            4 a (u^4) + 12 (a^2) (u^4) - 24 (a^3) (u^4) + 42 (a^4) (u^4) - 
            56 (a^5) (u^4) + 68 (a^6) (u^4) - 64 (a^7) (u^4) + 
            57 (a^8) (u^4) - 36 (a^9) (u^4) + 24 (a^10) (u^4) - 
            8 (a^11) (u^4) + 4 (a^12) (u^4)) 
        (1 + 8 a + 34 (a^2) + 88 (a^3) + 139 (a^4) + 124 (a^5) + 
            26 (a^6) - 12 (a^7) + 41 (a^8) - 12 (a^9) + 4 (a^10) - 4 u - 
            12 a u - 24 (a^2) u - 4 (a^3) u + 44 (a^4) u + 40 (a^5) u + 
            140 (a^6) u + 52 (a^7) u + 72 (a^8) u + 28 (a^9) u - 
            4 (a^10) u + 8 (a^11) u + 6 (u^2) - 4 a (u^2) + 
            6 (a^2) (u^2) - 36 (a^3) (u^2) + 4 (a^4) (u^2) - 28 (a^5) (u^2) + 
            26 (a^6) (u^2) + 44 (a^7) (u^2) + 38 (a^8) (u^2) + 
            48 (a^9) (u^2) + 20 (a^10) (u^2) + 8 (a^11) (u^2) + 
            4 (a^12) (u^2) - 4 (u^3) + 12 a (u^3) - 28 (a^2) (u^3) + 
            48 (a^3) (u^3) - 64 (a^4) (u^3) + 72 (a^5) (u^3) - 
            56 (a^6) (u^3) + 48 (a^7) (u^3) - 4 (a^8) (u^3) + 
            12 (a^9) (u^3) + 20 (a^10) (u^3) + 8 (a^12) (u^3) + (u^4) - 
            4 a (u^4) + 12 (a^2) (u^4) - 24 (a^3) (u^4) + 42 (a^4) (u^4) - 
            56 (a^5) (u^4) + 68 (a^6) (u^4) - 64 (a^7) (u^4) + 
            57 (a^8) (u^4) - 36 (a^9) (u^4) + 24 (a^10) (u^4) - 
            8 (a^11) (u^4) + 4 (a^12) (u^4)) 
        (1 + 6 a + 20 (a^2) + 36 (a^3) + 31 (a^4) + 22 (a^5) + 56 (a^6) + 
            184 (a^7) + 117 (a^8) - 118 (a^9) + 110 (a^10) - 32 (a^11) + 
            8 (a^12) - 4 u - 12 a u - 28 (a^2) u - 16 (a^3) u + 
            20 (a^4) u + 36 (a^5) u + 184 (a^6) u + 92 (a^7) u + 
            212 (a^8) u + 80 (a^9) u + 68 (a^10) u + 36 (a^11) u - 
            4 (a^12) u + 8 (a^13) u + 6 (u^2) + 18 (a^2) (u^2) - 
            24 (a^3) (u^2) + 8 (a^4) (u^2) - 44 (a^5) (u^2) + 
            106 (a^6) (u^2) - 64 (a^7) (u^2) + 282 (a^8) (u^2) - 
            92 (a^9) (u^2) + 228 (a^10) (u^2) - 48 (a^11) (u^2) + 
            60 (a^12) (u^2) + 4 (a^14) (u^2) - 4 (u^3) + 12 a (u^3) - 
            32 (a^2) (u^3) + 60 (a^3) (u^3) - 92 (a^4) (u^3) + 
            120 (a^5) (u^3) - 120 (a^6) (u^3) + 120 (a^7) (u^3) - 
            60 (a^8) (u^3) + 60 (a^9) (u^3) + 16 (a^10) (u^3) + 
            12 (a^11) (u^3) + 28 (a^12) (u^3) + 8 (a^14) (u^3) + (u^4) - 
            6 a (u^4) + 22 (a^2) (u^4) - 56 (a^3) (u^4) + 114 (a^4) (u^4) - 
            188 (a^5) (u^4) + 264 (a^6) (u^4) - 312 (a^7) (u^4) + 
            321 (a^8) (u^4) - 278 (a^9) (u^4) + 210 (a^10) (u^4) - 
            128 (a^11) (u^4) + 68 (a^12) (u^4) - 24 (a^13) (u^4) + 
            8 (a^14) (u^4));

\({x\_1, y\_1} = {
        \((\((1 + a\^2)\)\ 
              \((1 + 8\ a + 34\ a\^2 + 88\ a\^3 + 139\ a\^4 + 124\ a\^5 + 
                  26\ a\^6 - 12\ a\^7 + 41\ a\^8 - 12\ a\^9 + 4\ a\^10 - 
                  4\ u - 20\ a\ u - 44\ a\^2\ u - 32\ a\^3\ u + 80\ a\^4\ u + 
                  160\ a\^5\ u + 140\ a\^6\ u + 172\ a\^7\ u + 36\ a\^8\ u + 
                  16\ a\^10\ u + 6\ u\^2 + 12\ a\ u\^2 - 2\ a\^2\ u\^2 - 
                  20\ a\^3\ u\^2 - 68\ a\^4\ u\^2 + 20\ a\^5\ u\^2 + 
                  114\ a\^6\ u\^2 - 4\ a\^7\ u\^2 + 230\ a\^8\ u\^2 - 
                  56\ a\^9\ u\^2 + 56\ a\^10\ u\^2 - 4\ u\^3 + 4\ a\ u\^3 - 
                  12\ a\^3\ u\^3 + 32\ a\^4\ u\^3 - 44\ a\^5\ u\^3 + 
                  56\ a\^6\ u\^3 - 20\ a\^7\ u\^3 + 36\ a\^8\ u\^3 + 
                  24\ a\^9\ u\^3 + 8\ a\^10\ u\^3 + 16\ a\^11\ u\^3 + u\^4 - 
                  4\ a\ u\^4 + 12\ a\^2\ u\^4 - 24\ a\^3\ u\^4 + 
                  42\ a\^4\ u\^4 - 56\ a\^5\ u\^4 + 68\ a\^6\ u\^4 - 
                  64\ a\^7\ u\^4 + 57\ a\^8\ u\^4 - 36\ a\^9\ u\^4 + 
                  24\ a\^10\ u\^4 - 8\ a\^11\ u\^4 + 4\ a\^12\ u\^4)\))\)/
          \((\((\(-2\) + a)\)\ a\ 
              \((\(-1\) - 4\ a - 9\ a\^2 - 8\ a\^3 + 3\ a\^4 - 2\ a\^5 + 
                    u\^2 - 2\ a\ u\^2 + 4\ a\^2\ u\^2 - 4\ a\^3\ u\^2 + 
                    5\ a\^4\ u\^2 - 2\ a\^5\ u\^2 + 2\ a\^6\ u\^2)\)^2)\), 
        \((1 + 2\ a)\)/
          \((\((\(-2\) + a)\)\^2\ a\^2\ 
              \((\(-1\) - 4\ a - 9\ a\^2 - 8\ a\^3 + 3\ a\^4 - 2\ a\^5 + 
                    u\^2 - 2\ a\ u\^2 + 4\ a\^2\ u\^2 - 4\ a\^3\ u\^2 + 
                    5\ a\^4\ u\^2 - 2\ a\^5\ u\^2 + 2\ a\^6\ u\^2)\)^3)\)}; 
    
{x\_2, y\_2} = {
        \((\((\(-2\) + a)\)\ a\ \((1 + a\^2)\)\ 
              \((1 + 8\ a + 34\ a\^2 + 88\ a\^3 + 139\ a\^4 + 124\ a\^5 + 
                  26\ a\^6 - 12\ a\^7 + 41\ a\^8 - 12\ a\^9 + 4\ a\^10 - 
                  4\ u - 20\ a\ u - 44\ a\^2\ u - 32\ a\^3\ u + 80\ a\^4\ u + 
                  160\ a\^5\ u + 140\ a\^6\ u + 172\ a\^7\ u + 36\ a\^8\ u + 
                  16\ a\^10\ u + 6\ u\^2 + 12\ a\ u\^2 - 2\ a\^2\ u\^2 - 
                  20\ a\^3\ u\^2 - 68\ a\^4\ u\^2 + 20\ a\^5\ u\^2 + 
                  114\ a\^6\ u\^2 - 4\ a\^7\ u\^2 + 230\ a\^8\ u\^2 - 
                  56\ a\^9\ u\^2 + 56\ a\^10\ u\^2 - 4\ u\^3 + 4\ a\ u\^3 - 
                  12\ a\^3\ u\^3 + 32\ a\^4\ u\^3 - 44\ a\^5\ u\^3 + 
                  56\ a\^6\ u\^3 - 20\ a\^7\ u\^3 + 36\ a\^8\ u\^3 + 
                  24\ a\^9\ u\^3 + 8\ a\^10\ u\^3 + 16\ a\^11\ u\^3 + u\^4 - 
                  4\ a\ u\^4 + 12\ a\^2\ u\^4 - 24\ a\^3\ u\^4 + 
                  42\ a\^4\ u\^4 - 56\ a\^5\ u\^4 + 68\ a\^6\ u\^4 - 
                  64\ a\^7\ u\^4 + 57\ a\^8\ u\^4 - 36\ a\^9\ u\^4 + 
                  24\ a\^10\ u\^4 - 8\ a\^11\ u\^4 + 4\ a\^12\ u\^4)\))\)/
          \((\(-1\) - 3\ a - 4\ a\^2 + 5\ a\^3 + 20\ a\^4 + 3\ a\^5 - a\^6 + 
                2\ a\^7 + 2\ u + 8\ a\^3\ u - 6\ a\^4\ u + 16\ a\^5\ u - 
                4\ a\^6\ u + 8\ a\^7\ u - u\^2 + 3\ a\ u\^2 - 5\ a\^2\ u\^2 + 
                6\ a\^3\ u\^2 - 5\ a\^4\ u\^2 + 3\ a\^5\ u\^2 + a\^6\ u\^2 + 
                2\ a\^8\ u\^2)\)^2, 
        \((\((\(-2\) + a)\)\ a\ \((1 + 2\ a)\))\)/
          \((\(-1\) - 3\ a - 4\ a\^2 + 5\ a\^3 + 20\ a\^4 + 3\ a\^5 - a\^6 + 
                2\ a\^7 + 2\ u + 8\ a\^3\ u - 6\ a\^4\ u + 16\ a\^5\ u - 
                4\ a\^6\ u + 8\ a\^7\ u - u\^2 + 3\ a\ u\^2 - 5\ a\^2\ u\^2 + 
                6\ a\^3\ u\^2 - 5\ a\^4\ u\^2 + 3\ a\^5\ u\^2 + a\^6\ u\^2 + 
                2\ a\^8\ u\^2)\)^3}; 
    
{x\_3, y\_3} = {
        \((\((\(-2\) + a)\)\^2\ a\^2\ 
              \((1 + 8\ a + 34\ a\^2 + 88\ a\^3 + 139\ a\^4 + 124\ a\^5 + 
                  26\ a\^6 - 12\ a\^7 + 41\ a\^8 - 12\ a\^9 + 4\ a\^10 - 
                  4\ u - 12\ a\ u - 24\ a\^2\ u - 4\ a\^3\ u + 44\ a\^4\ u + 
                  40\ a\^5\ u + 140\ a\^6\ u + 52\ a\^7\ u + 72\ a\^8\ u + 
                  28\ a\^9\ u - 4\ a\^10\ u + 8\ a\^11\ u + 6\ u\^2 - 
                  4\ a\ u\^2 + 6\ a\^2\ u\^2 - 36\ a\^3\ u\^2 + 
                  4\ a\^4\ u\^2 - 28\ a\^5\ u\^2 + 26\ a\^6\ u\^2 + 
                  44\ a\^7\ u\^2 + 38\ a\^8\ u\^2 + 48\ a\^9\ u\^2 + 
                  20\ a\^10\ u\^2 + 8\ a\^11\ u\^2 + 4\ a\^12\ u\^2 - 
                  4\ u\^3 + 12\ a\ u\^3 - 28\ a\^2\ u\^3 + 48\ a\^3\ u\^3 - 
                  64\ a\^4\ u\^3 + 72\ a\^5\ u\^3 - 56\ a\^6\ u\^3 + 
                  48\ a\^7\ u\^3 - 4\ a\^8\ u\^3 + 12\ a\^9\ u\^3 + 
                  20\ a\^10\ u\^3 + 8\ a\^12\ u\^3 + u\^4 - 4\ a\ u\^4 + 
                  12\ a\^2\ u\^4 - 24\ a\^3\ u\^4 + 42\ a\^4\ u\^4 - 
                  56\ a\^5\ u\^4 + 68\ a\^6\ u\^4 - 64\ a\^7\ u\^4 + 
                  57\ a\^8\ u\^4 - 36\ a\^9\ u\^4 + 24\ a\^10\ u\^4 - 
                  8\ a\^11\ u\^4 + 4\ a\^12\ u\^4)\))\)/
          \((\((1 + a\^2)\)\^2\ 
              \((1 + 4\ a + 9\ a\^2 + 8\ a\^3 - 3\ a\^4 + 2\ a\^5 - 2\ u - 
                    2\ a\ u - 4\ a\^2\ u + 18\ a\^3\ u + 4\ a\^4\ u - 
                    2\ a\^5\ u + 2\ a\^6\ u + u\^2 - 2\ a\ u\^2 + 
                    4\ a\^2\ u\^2 - 4\ a\^3\ u\^2 + 5\ a\^4\ u\^2 - 
                    2\ a\^5\ u\^2 + 2\ a\^6\ u\^2)\)^2)\), 
        \(-\(\((\((\(-2\) + a)\)\ a\ \((1 + 2\ a)\))\)/
            \((\((1 + a\^2)\)\^3\ 
                \((1 + 4\ a + 9\ a\^2 + 8\ a\^3 - 3\ a\^4 + 2\ a\^5 - 2\ u - 
                      2\ a\ u - 4\ a\^2\ u + 18\ a\^3\ u + 4\ a\^4\ u - 
                      2\ a\^5\ u + 2\ a\^6\ u + u\^2 - 2\ a\ u\^2 + 
                      4\ a\^2\ u\^2 - 4\ a\^3\ u\^2 + 5\ a\^4\ u\^2 - 
                      2\ a\^5\ u\^2 + 2\ a\^6\ u\^2)\)^3)\)\)\)}; \)

Factor[g ((Subscript[y, 1])^2) - (f /. x -> (Subscript[x, 1]))]

Factor[g ((Subscript[y, 2])^2) - (f /. x -> (Subscript[x, 2]))]

Factor[g ((Subscript[y, 3])^2) - (f /. x -> (Subscript[x, 3]))]

Theorem 5.3

f = x (x - b) (x - (a^2) b);

t  in terms of  u

t = 
      ((a^2) b 
            (1 + 2 (a^2) + (a^4) - 4 a u + 8 (a^2) u - 8 (a^3) u + 
                8 (a^4) u - 4 (a^5) u + 4 (u^2) - 8 a (u^2) + 
                14 (a^2) (u^2) - 32 (a^3) (u^2) + 40 (a^4) (u^2) - 
                24 (a^5) (u^2) + 6 (a^6) (u^2) + 16 (a^2) (u^3) - 
                60 (a^3) (u^3) + 88 (a^4) (u^3) - 64 (a^5) (u^3) + 
                24 (a^6) (u^3) - 4 (a^7) (u^3) + 16 (a^2) (u^4) - 
                56 (a^3) (u^4) + 81 (a^4) (u^4) - 64 (a^5) (u^4) + 
                30 (a^6) (u^4) - 8 (a^7) (u^4) + (a^8) (u^4)))/
        (((-1 - (a^2) - 4 a (u^2) + 7 (a^2) (u^2) - 4 (a^3) (u^2) + 
              (a^4) (u^2)))^2);

g = 
      Factor[(f /. x -> t) 
          (((((((-1 - (a^2) - 4 a (u^2) + 7 (a^2) (u^2) - 4 (a^3) (u^2) + 
                      (a^4) (u^2)))^3)/((-1 + a) (a^2) b))))^2)]

\({x\_1, y\_1} = {
        \((a\^2\ b\ 
              \((1 + 2\ a\^2 + a\^4 - 4\ a\ u + 8\ a\^2\ u - 8\ a\^3\ u + 
                  8\ a\^4\ u - 4\ a\^5\ u + 4\ u\^2 - 8\ a\ u\^2 + 
                  14\ a\^2\ u\^2 - 32\ a\^3\ u\^2 + 40\ a\^4\ u\^2 - 
                  24\ a\^5\ u\^2 + 6\ a\^6\ u\^2 + 16\ a\^2\ u\^3 - 
                  60\ a\^3\ u\^3 + 88\ a\^4\ u\^3 - 64\ a\^5\ u\^3 + 
                  24\ a\^6\ u\^3 - 4\ a\^7\ u\^3 + 16\ a\^2\ u\^4 - 
                  56\ a\^3\ u\^4 + 81\ a\^4\ u\^4 - 64\ a\^5\ u\^4 + 
                  30\ a\^6\ u\^4 - 8\ a\^7\ u\^4 + a\^8\ u\^4)\))\)/
          \((\(-1\) - a\^2 - 4\ a\ u\^2 + 7\ a\^2\ u\^2 - 4\ a\^3\ u\^2 + 
                a\^4\ u\^2)\)\^2, 
        \(\((\(-1\) + a)\)\ a\^2\ b
            \)\/\((\(-1\) - a\^2 - 4\ a\ u\^2 + 7\ a\^2\ u\^2 - 
                4\ a\^3\ u\^2 + a\^4\ u\^2)\)\^3}; 
    
{x\_2, y\_2} = {
        \((a\^2\ b\ 
              \((1 + 2\ a\^2 + a\^4 - 4\ a\ u + 8\ a\^2\ u - 8\ a\^3\ u + 
                  8\ a\^4\ u - 4\ a\^5\ u + 4\ u\^2 - 8\ a\ u\^2 + 
                  14\ a\^2\ u\^2 - 32\ a\^3\ u\^2 + 40\ a\^4\ u\^2 - 
                  24\ a\^5\ u\^2 + 6\ a\^6\ u\^2 + 16\ a\^2\ u\^3 - 
                  60\ a\^3\ u\^3 + 88\ a\^4\ u\^3 - 64\ a\^5\ u\^3 + 
                  24\ a\^6\ u\^3 - 4\ a\^7\ u\^3 + 16\ a\^2\ u\^4 - 
                  56\ a\^3\ u\^4 + 81\ a\^4\ u\^4 - 64\ a\^5\ u\^4 + 
                  30\ a\^6\ u\^4 - 8\ a\^7\ u\^4 + a\^8\ u\^4)\))\)/
          \((a + a\^3 - 2\ u + 4\ a\ u - 4\ a\^2\ u + 4\ a\^3\ u - 
                2\ a\^4\ u - 4\ a\^2\ u\^2 + 7\ a\^3\ u\^2 - 4\ a\^4\ u\^2 + 
                a\^5\ u\^2)\)^2, 
        \(-\(\((\((\(-1\) + a)\)\ a\^2\ b)\)/
            \((a + a\^3 - 2\ u + 4\ a\ u - 4\ a\^2\ u + 4\ a\^3\ u - 
                  2\ a\^4\ u - 4\ a\^2\ u\^2 + 7\ a\^3\ u\^2 - 
                  4\ a\^4\ u\^2 + a\^5\ u\^2)\)^3\)\)}; 
    
{x\_3, y\_3} = {
        \(-\(\((16\ a\^2\ b\ u\ \((\(-a\) + u - 2\ a\ u + a\^2\ u)\)\ 
                \((1 - a + a\^2 - a\^3 + 4\ a\^2\ u - 3\ a\^3\ u + a\^4\ u)
                  \)\ \((1 + 2\ a\^2 + a\^4 - 4\ a\ u + 8\ a\^2\ u - 
                    8\ a\^3\ u + 8\ a\^4\ u - 4\ a\^5\ u + 4\ u\^2 - 
                    8\ a\ u\^2 + 14\ a\^2\ u\^2 - 32\ a\^3\ u\^2 + 
                    40\ a\^4\ u\^2 - 24\ a\^5\ u\^2 + 6\ a\^6\ u\^2 + 
                    16\ a\^2\ u\^3 - 60\ a\^3\ u\^3 + 88\ a\^4\ u\^3 - 
                    64\ a\^5\ u\^3 + 24\ a\^6\ u\^3 - 4\ a\^7\ u\^3 + 
                    16\ a\^2\ u\^4 - 56\ a\^3\ u\^4 + 81\ a\^4\ u\^4 - 
                    64\ a\^5\ u\^4 + 30\ a\^6\ u\^4 - 8\ a\^7\ u\^4 + 
                    a\^8\ u\^4)\))\)/
            \((\((1 + a + 2\ a\ u - 2\ a\^2\ u + 4\ a\ u\^2 - 3\ a\^2\ u\^2 + 
                    a\^3\ u\^2)\)\ 
                \((\(-1\) + a - a\^2 + a\^3 - 8\ a\^2\ u + 6\ a\^3\ u - 
                      2\ a\^4\ u + 4\ a\ u\^2 - 11\ a\^2\ u\^2 + 
                      11\ a\^3\ u\^2 - 5\ a\^4\ u\^2 + a\^5\ u\^2)\)^2\ 
                \((1 + 2\ a\^2 + a\^4 - 2\ a\ u + 4\ a\^2\ u - 4\ a\^3\ u + 
                    4\ a\^4\ u - 2\ a\^5\ u + 4\ a\ u\^2 - 7\ a\^2\ u\^2 + 
                    6\ a\^4\ u\^2 - 4\ a\^5\ u\^2 + a\^6\ u\^2)\))\)\)\), 
        \(-\(\((2\ a\^2\ \((1 + a)\)\ b\ 
                \((\(-1\) + a - 2\ a\^2 + 2\ a\^3 - a\^4 + a\^5 + 8\ a\ u - 
                    16\ a\^2\ u + 20\ a\^3\ u - 20\ a\^4\ u + 12\ a\^5\ u - 
                    4\ a\^6\ u - 8\ u\^2 + 24\ a\ u\^2 - 38\ a\^2\ u\^2 + 
                    78\ a\^3\ u\^2 - 96\ a\^4\ u\^2 + 72\ a\^5\ u\^2 - 
                    30\ a\^6\ u\^2 + 6\ a\^7\ u\^2 - 32\ a\^2\ u\^3 + 
                    120\ a\^3\ u\^3 - 192\ a\^4\ u\^3 + 172\ a\^5\ u\^3 - 
                    92\ a\^6\ u\^3 + 28\ a\^7\ u\^3 - 4\ a\^8\ u\^3 - 
                    16\ a\^2\ u\^4 + 72\ a\^3\ u\^4 - 137\ a\^4\ u\^4 + 
                    145\ a\^5\ u\^4 - 94\ a\^6\ u\^4 + 38\ a\^7\ u\^4 - 
                    9\ a\^8\ u\^4 + a\^9\ u\^4)\)\ 
                \((\(-1\) + a - 2\ a\^2 + 2\ a\^3 - a\^4 + a\^5 - 
                    8\ a\^2\ u + 12\ a\^3\ u - 12\ a\^4\ u + 12\ a\^5\ u - 
                    4\ a\^6\ u - 6\ a\^2\ u\^2 + 14\ a\^3\ u\^2 - 
                    48\ a\^4\ u\^2 + 56\ a\^5\ u\^2 - 30\ a\^6\ u\^2 + 
                    6\ a\^7\ u\^2 + 32\ a\^3\ u\^3 - 104\ a\^4\ u\^3 + 
                    132\ a\^5\ u\^3 - 84\ a\^6\ u\^3 + 28\ a\^7\ u\^3 - 
                    4\ a\^8\ u\^3 - 16\ a\^2\ u\^4 + 72\ a\^3\ u\^4 - 
                    137\ a\^4\ u\^4 + 145\ a\^5\ u\^4 - 94\ a\^6\ u\^4 + 
                    38\ a\^7\ u\^4 - 9\ a\^8\ u\^4 + a\^9\ u\^4)\))\)/
            \((\((1 + a + 2\ a\ u - 2\ a\^2\ u + 4\ a\ u\^2 - 3\ a\^2\ u\^2 + 
                      a\^3\ u\^2)\)\^2\ 
                \((\(-1\) + a - a\^2 + a\^3 - 8\ a\^2\ u + 6\ a\^3\ u - 
                      2\ a\^4\ u + 4\ a\ u\^2 - 11\ a\^2\ u\^2 + 
                      11\ a\^3\ u\^2 - 5\ a\^4\ u\^2 + a\^5\ u\^2)\)^3\ 
                \((1 + 2\ a\^2 + a\^4 - 2\ a\ u + 4\ a\^2\ u - 4\ a\^3\ u + 
                      4\ a\^4\ u - 2\ a\^5\ u + 4\ a\ u\^2 - 7\ a\^2\ u\^2 + 
                      6\ a\^4\ u\^2 - 4\ a\^5\ u\^2 + a\^6\ u\^2)\)^2)
              \)\)\)}; \)

Factor[g ((Subscript[y, 1])^2) - (f /. x -> (Subscript[x, 1]))]

Factor[g ((Subscript[y, 2])^2) - (f /. x -> (Subscript[x, 2]))]

Factor[g ((Subscript[y, 3])^2) - (f /. x -> (Subscript[x, 3]))]

Theorem 5.5

\(f = x\^3 - x; 
g = 6 \((u\^12 - 33  u\^8 - 33  u\^4 + 1)\); 
    
{x\_1, y\_1} = {
        \(-\(\(u\^4 - 6  u\^2 + 1\)\/\(3\ \((u\^2 + 1)\)\^2\)\)\), 
        2\/\(9\ \((u\^2 + 1)\)\^3\)}; 
    
{x\_2, y\_2} = {
        \(-\(\(u\^4 + 6  u\^2 + 1\)\/\(3\ \((u\^2 - 1)\)\^2\)\)\), 
        \(2\ \)\/\(9\ \((u\^2 - 1)\)\^3\)}; 
    
{x\_3, y\_3} = {\(u\^4 + 1\)\/\(6\ u\^2\), 1\/\(36\ u\^3\)}; \)

Factor[g ((Subscript[y, 1])^2) - (f /. x -> (Subscript[x, 1]))]

Factor[g ((Subscript[y, 2])^2) - (f /. x -> (Subscript[x, 2]))]

Factor[g ((Subscript[y, 3])^2) - (f /. x -> (Subscript[x, 3]))]