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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 32871, 804]*) (*NotebookOutlinePosition[ 57033, 1676]*) (* CellTagsIndexPosition[ 56989, 1672]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData[{ StyleBox["This ", FontSize->18, FontWeight->"Plain", FontColor->RGBColor[0, 0, 1]], StyleBox["Mathematica", FontSize->18, FontWeight->"Plain", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox[" notebook contains the formulas to accompany \"", FontSize->18, FontWeight->"Plain", FontColor->RGBColor[0, 0, 1]], StyleBox["Rank frequencies for quadratic twists of elliptic curves", FontSlant->"Italic", FontColor->RGBColor[0, 0, 1]], StyleBox["\", by K. Rubin and A. Silverberg.", FontSize->18, FontWeight->"Plain", FontColor->RGBColor[0, 0, 1]] }], "Section", CellDingbat->None], Cell[TextData[{ "In each example, the points ", Cell[BoxData[ \(TraditionalForm\`\(\((x\_i, y\_i)\)\ \)\)]], "are independent points of infinite order on the elliptic curve ", Cell[BoxData[ \(TraditionalForm\`\(g(u)\) y\^2 = \ f(x)\)]], "." }], "Section", CellDingbat->None], Cell[CellGroupData[{ Cell["Rank 2 examples", "Section"], Cell[CellGroupData[{ Cell["Corollary 4.2", "Subsection"], Cell[BoxData[ \(f = x\^3 + a\ x\^2 + b\ x; \n g = \(-a\)\ b\ \((b\^2 + u\^2)\)\ \((u\^4 + \((2\ b\^2 - a\^2\ b)\)\ u\^2 + b\^4)\); \n{x\_1, y\_1} = {\(-\(\(b\^2 + u\^2\)\/\(a\ b\)\)\), 1\/\(a\^2\ b\^2\)}; \n{x\_2, y\_2} = {\(-\(\(b\ \((b\^2 + u\^2)\)\)\/\(a\ u\^2\)\)\), b\/\(a\^2\ u\^3\)}; \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[g\ y\_1\^2 - \((f /. x -> x\_1)\)]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[g\ y\_2\^2 - \((f /. x -> x\_2)\)]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Corollary 4.4", "Subsection"], Cell[BoxData[ \(f = x\^3 + b\^2\/\(4 c\)\ x\^2 + b\ x + c; \n 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)\); \n{x\_1, y\_1} = {\(-\(\(3\ c\^2 + u\^2\)\/\(2\ b\ c\)\)\), 1\/\(4\ b\^2\ c\^2\)}; \n{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)\)}; \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[g\ y\_1\^2 - \((f /. x -> x\_1)\)]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[g\ y\_2\^2 - \((f /. x -> x\_2)\)]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Theorem 4.6", "Subsection"], Cell[BoxData[ \(f = x\^3 + a\ x + b; \n g = \(-a\)\ b \((\(b\^2\) \((u\^4 + u\^2 + 1)\)\^3 + \(a\^3\) \(u\^4\) \((u\^2 + 1)\)\^2)\) \((u\^2 + 1)\); \n{x\_1, y\_1} = { \(-\(\(b\ \((\(-1\) + u\^6)\)\)\/\(a\ \((\(-1\) + u\^4)\)\)\)\), 1\/\(a\^2\ \((1 + u\^2)\)\^2\)}; \n{x\_2, y\_2} = { \(-\(\(b\ \((\(-1\) + u\^6)\)\)\/\(a\ u\^2\ \((\(-1\) + u\^4)\)\)\)\), 1\/\(a\^2\ u\^3\ \((1 + u\^2)\)\^2\)}; \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[g\ y\_1\^2 - \((f /. x -> x\_1)\)]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Simplify[g\ y\_2\^2 - \((f /. x -> x\_2)\)]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]] }, Closed]] }, Open ]], Cell[" ", "Section", CellDingbat->None], Cell[CellGroupData[{ Cell["Rank 3 examples", "Section"], Cell[CellGroupData[{ Cell["Theorem 5.1", "Subsection"], Cell[BoxData[ \(\[Lambda] = \(-2\) a\^2; \nf = x \((x - 1)\) \((x - \[Lambda])\); \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(d = \[Lambda] \((2 \[Lambda] - 1)\) u\^2 + 2 - \[Lambda]; 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\n{x\_2, y\_2} = { \(\(\[Lambda]\^2\) \((d\^2 - 4 \[Lambda]\ u \((u - 1)\) \((\[Lambda] \((2 \[Lambda] - 1)\) u + 2 - \[Lambda])\)) \)\)\/\(( \[Lambda] \((2 \[Lambda] - 1)\) u\^2 - 2 \[Lambda] \((2 \[Lambda] - 1)\) u + \[Lambda] - 2)\)\^2, \(a\ \[Lambda]\)\/\(( \[Lambda] \((2 \[Lambda] - 1)\) u\^2 - 2 \[Lambda] \((2 \[Lambda] - 1)\) u + \[Lambda] - 2)\)\^3}; \n{x\_3, y\_3} = { \(d\^2 + 4 \[Lambda]\ u \((u - 1)\) \((\[Lambda] \((2 \[Lambda] - 1)\) u + 2 - \[Lambda]) \)\)\/\(\[Lambda] \((\[Lambda] \((2 \[Lambda] - 1)\) u\^2 - \((2 \[Lambda] - 4)\) u + \[Lambda] - 2)\)\^2\), \(-\(a\/\(\(\[Lambda]\^2\) \((\[Lambda] \((2 \[Lambda] - 1)\) u\^2 - \((2 \[Lambda] - 4)\) u + \[Lambda] - 2)\)\^3\)\)\)}; \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Factor[g\ y\_1\^2 - \((f /. x -> x\_1)\)]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Factor[g\ y\_2\^2 - \((f /. x -> x\_2)\)]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Factor[g\ y\_3\^2 - \((f /. x -> x\_3)\)]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Theorem 5.2(a)", "Subsection"], Cell[BoxData[ \(\[Lambda] = \(1 - a\^2\)\/\(a\^2 + 2\); \n f = x \((x - 1)\) \((x - \[Lambda])\); \)], "Input"], Cell[BoxData[ \(\(g = \(-2\)\ \((1 + \[Lambda])\)\ \((4\ u\^2 + 8\ u\^3 + 4\ u\^4 - 4\ u\ \[Lambda] - 16\ u\^2\ \[Lambda] - 20\ u\^3\ \[Lambda] - 4\ u\^4\ \[Lambda] + \[Lambda]\^2 + 12\ u\ \[Lambda]\^2 + 26\ u\^2\ \[Lambda]\^2 + 8\ u\^3\ \[Lambda]\^2 + u\^4\ \[Lambda]\^2 - 2\ \[Lambda]\^3 - 12\ u\ \[Lambda]\^3 - 10\ u\^2\ \[Lambda]\^3 + 3\ \[Lambda]\^4 + 8\ u\ \[Lambda]\^4 - 2\ u\^2\ \[Lambda]\^4 - 2\ \[Lambda]\^5 + \[Lambda]\^6)\)\ \((4\ u\^4 - 8\ u\^3\ \[Lambda] - 4\ u\^4\ \[Lambda] + \[Lambda]\^2 + 4\ u\ \[Lambda]\^2 + 14\ u\^2\ \[Lambda]\^2 + 20\ u\^3\ \[Lambda]\^2 + u\^4\ \[Lambda]\^2 - 2\ \[Lambda]\^3 - 12\ u\ \[Lambda]\^3 - 26\ u\^2\ \[Lambda]\^3 - 8\ u\^3\ \[Lambda]\^3 + 3\ \[Lambda]\^4 + 12\ u\ \[Lambda]\^4 + 14\ u\^2\ \[Lambda]\^4 - 2\ \[Lambda]\^5 - 8\ u\ \[Lambda]\^5 + \[Lambda]\^6)\)\ \((2\ u\^2 + 4\ u\^3 + 4\ u\^4 - 2\ u\ \[Lambda] - 8\ u\^2\ \[Lambda] - 14\ u\^3\ \[Lambda] - 4\ u\^4\ \[Lambda] + \[Lambda]\^2 + 8\ u\ \[Lambda]\^2 + 20\ u\^2\ \[Lambda]\^2 + 14\ u\^3\ \[Lambda]\^2 + u\^4\ \[Lambda]\^2 - 2\ \[Lambda]\^3 - 12\ u\ \[Lambda]\^3 - 18\ u\^2\ \[Lambda]\^3 - 4\ u\^3\ \[Lambda]\^3 + 3\ \[Lambda]\^4 + 10\ u\ \[Lambda]\^4 + 6\ u\^2\ \[Lambda]\^4 - 2\ \[Lambda]\^5 - 4\ u\ \[Lambda]\^5 + \[Lambda]\^6)\); \)\)], "Input"], Cell[BoxData[ \({x\_1, y\_1} = { \((2\ \[Lambda]\ \((2\ u\^2 + 4\ u\^3 + 4\ u\^4 - 2\ u\ \[Lambda] - 8\ u\^2\ \[Lambda] - 14\ u\^3\ \[Lambda] - 4\ u\^4\ \[Lambda] + \[Lambda]\^2 + 8\ u\ \[Lambda]\^2 + 20\ u\^2\ \[Lambda]\^2 + 14\ u\^3\ \[Lambda]\^2 + u\^4\ \[Lambda]\^2 - 2\ \[Lambda]\^3 - 12\ u\ \[Lambda]\^3 - 18\ u\^2\ \[Lambda]\^3 - 4\ u\^3\ \[Lambda]\^3 + 3\ \[Lambda]\^4 + 10\ u\ \[Lambda]\^4 + 6\ u\^2\ \[Lambda]\^4 - 2\ \[Lambda]\^5 - 4\ u\ \[Lambda]\^5 + \[Lambda]\^6)\))\)/ \((\((1 + \[Lambda])\)\ \((2\ u\^2 + \[Lambda] - u\^2\ \[Lambda] - \[Lambda]\^2 + \[Lambda]\^3)\)\^2)\), \(-\(\(\((\(-1\) + \[Lambda])\)\ \[Lambda]\)\/\(\((1 + \[Lambda])\)\^2\ \((2\ u\^2 + \[Lambda] - u\^2\ \[Lambda] - \[Lambda]\^2 + \[Lambda]\^3)\)\^3\)\)\)}; \n{x\_2, y\_2} = { \((\[Lambda]\ \((4\ u\^2 + 8\ u\^3 + 4\ u\^4 - 4\ u\ \[Lambda] - 16\ u\^2\ \[Lambda] - 20\ u\^3\ \[Lambda] - 4\ u\^4\ \[Lambda] + \[Lambda]\^2 + 12\ u\ \[Lambda]\^2 + 26\ u\^2\ \[Lambda]\^2 + 8\ u\^3\ \[Lambda]\^2 + u\^4\ \[Lambda]\^2 - 2\ \[Lambda]\^3 - 12\ u\ \[Lambda]\^3 - 10\ u\^2\ \[Lambda]\^3 + 3\ \[Lambda]\^4 + 8\ u\ \[Lambda]\^4 - 2\ u\^2\ \[Lambda]\^4 - 2\ \[Lambda]\^5 + \[Lambda]\^6)\)) \)/\((\((\(-1\) + 2\ \[Lambda])\)\ \((\(-2\)\ u\^2 + \[Lambda] + 4\ u\ \[Lambda] + u\^2\ \[Lambda] - \[Lambda]\^2 - 2\ u\ \[Lambda]\^2 + \[Lambda]\^3)\)\^2)\), \(a\ \((\(-1\) + \[Lambda])\)\ \[Lambda]\)\/\(\((\(-1\) + 2\ \[Lambda])\)\^2\ \((\(-2\)\ u\^2 + \[Lambda] + 4\ u\ \[Lambda] + u\^2\ \[Lambda] - \[Lambda]\^2 - 2\ u\ \[Lambda]\^2 + \[Lambda]\^3)\)\^3\)}; \n{x\_3, y\_3} = { \((\[Lambda]\ \((4\ u\^4 - 8\ u\^3\ \[Lambda] - 4\ u\^4\ \[Lambda] + \[Lambda]\^2 + 4\ u\ \[Lambda]\^2 + 14\ u\^2\ \[Lambda]\^2 + 20\ u\^3\ \[Lambda]\^2 + u\^4\ \[Lambda]\^2 - 2\ \[Lambda]\^3 - 12\ u\ \[Lambda]\^3 - 26\ u\^2\ \[Lambda]\^3 - 8\ u\^3\ \[Lambda]\^3 + 3\ \[Lambda]\^4 + 12\ u\ \[Lambda]\^4 + 14\ u\^2\ \[Lambda]\^4 - 2\ \[Lambda]\^5 - 8\ u\ \[Lambda]\^5 + \[Lambda]\^6)\))\)/ \((\((\(-1\) + 2\ \[Lambda])\)\ \((\(-2\)\ u - 2\ u\^2 + \[Lambda] + 2\ u\ \[Lambda] + u\^2\ \[Lambda] - \[Lambda]\^2 - 2\ u\ \[Lambda]\^2 + \[Lambda]\^3)\)\^2)\), \(-\(\(a\ \((\(-1\) + \[Lambda])\)\ \[Lambda]\)\/\(\((\(-1\) + 2\ \[Lambda])\)\^2\ \((\(-2\)\ u - 2\ u\^2 + \[Lambda] + 2\ u\ \[Lambda] + u\^2\ \[Lambda] - \[Lambda]\^2 - 2\ u\ \[Lambda]\^2 + \[Lambda]\^3)\)\^3\)\)\)}; \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Factor[g\ y\_1\^2 - \((f /. x -> x\_1)\)]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Factor[g\ y\_2\^2 - \((f /. x -> x\_2)\)]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Factor[g\ y\_3\^2 - \((f /. x -> x\_3)\)]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Theorem 5.2(b)", "Subsection"], Cell[BoxData[ \(\[Lambda] = \(a \((a - 2)\)\)\/\(a\^2 + 1\); \n f = x \((x - 1)\) \((x - \[Lambda])\); \)], "Input"], Cell[BoxData[ \(\(g = \((\(-2\) + a)\)\ a\ \((1 + 8\ a + 34\ a\^2 + 88\ a\^3 + 139\ a\^4 + 124\ a\^5 + 26\ a\^6 - 12\ a\^7 + 41\ a\^8 - 12\ a\^9 + 4\ a\^10 - 4\ u - 20\ a\ u - 44\ a\^2\ u - 32\ a\^3\ u + 80\ a\^4\ u + 160\ a\^5\ u + 140\ a\^6\ u + 172\ a\^7\ u + 36\ a\^8\ u + 16\ a\^10\ u + 6\ u\^2 + 12\ a\ u\^2 - 2\ a\^2\ u\^2 - 20\ a\^3\ u\^2 - 68\ a\^4\ u\^2 + 20\ a\^5\ u\^2 + 114\ a\^6\ u\^2 - 4\ a\^7\ u\^2 + 230\ a\^8\ u\^2 - 56\ a\^9\ u\^2 + 56\ a\^10\ u\^2 - 4\ u\^3 + 4\ a\ u\^3 - 12\ a\^3\ u\^3 + 32\ a\^4\ u\^3 - 44\ a\^5\ u\^3 + 56\ a\^6\ u\^3 - 20\ a\^7\ u\^3 + 36\ a\^8\ u\^3 + 24\ a\^9\ u\^3 + 8\ a\^10\ u\^3 + 16\ a\^11\ u\^3 + u\^4 - 4\ a\ u\^4 + 12\ a\^2\ u\^4 - 24\ a\^3\ u\^4 + 42\ a\^4\ u\^4 - 56\ a\^5\ u\^4 + 68\ a\^6\ u\^4 - 64\ a\^7\ u\^4 + 57\ a\^8\ u\^4 - 36\ a\^9\ u\^4 + 24\ a\^10\ u\^4 - 8\ a\^11\ u\^4 + 4\ a\^12\ u\^4)\)\ \((1 + 8\ a + 34\ a\^2 + 88\ a\^3 + 139\ a\^4 + 124\ a\^5 + 26\ a\^6 - 12\ a\^7 + 41\ a\^8 - 12\ a\^9 + 4\ a\^10 - 4\ u - 12\ a\ u - 24\ a\^2\ u - 4\ a\^3\ u + 44\ a\^4\ u + 40\ a\^5\ u + 140\ a\^6\ u + 52\ a\^7\ u + 72\ a\^8\ u + 28\ a\^9\ u - 4\ a\^10\ u + 8\ a\^11\ u + 6\ u\^2 - 4\ a\ u\^2 + 6\ a\^2\ u\^2 - 36\ a\^3\ u\^2 + 4\ a\^4\ u\^2 - 28\ a\^5\ u\^2 + 26\ a\^6\ u\^2 + 44\ a\^7\ u\^2 + 38\ a\^8\ u\^2 + 48\ a\^9\ u\^2 + 20\ a\^10\ u\^2 + 8\ a\^11\ u\^2 + 4\ a\^12\ u\^2 - 4\ u\^3 + 12\ a\ u\^3 - 28\ a\^2\ u\^3 + 48\ a\^3\ u\^3 - 64\ a\^4\ u\^3 + 72\ a\^5\ u\^3 - 56\ a\^6\ u\^3 + 48\ a\^7\ u\^3 - 4\ a\^8\ u\^3 + 12\ a\^9\ u\^3 + 20\ a\^10\ u\^3 + 8\ a\^12\ u\^3 + u\^4 - 4\ a\ u\^4 + 12\ a\^2\ u\^4 - 24\ a\^3\ u\^4 + 42\ a\^4\ u\^4 - 56\ a\^5\ u\^4 + 68\ a\^6\ u\^4 - 64\ a\^7\ u\^4 + 57\ a\^8\ u\^4 - 36\ a\^9\ u\^4 + 24\ a\^10\ u\^4 - 8\ a\^11\ u\^4 + 4\ a\^12\ u\^4)\)\ \((1 + 6\ a + 20\ a\^2 + 36\ a\^3 + 31\ a\^4 + 22\ a\^5 + 56\ a\^6 + 184\ a\^7 + 117\ a\^8 - 118\ a\^9 + 110\ a\^10 - 32\ a\^11 + 8\ a\^12 - 4\ u - 12\ a\ u - 28\ a\^2\ u - 16\ a\^3\ u + 20\ a\^4\ u + 36\ a\^5\ u + 184\ a\^6\ u + 92\ a\^7\ u + 212\ a\^8\ u + 80\ a\^9\ u + 68\ a\^10\ u + 36\ a\^11\ u - 4\ a\^12\ u + 8\ a\^13\ u + 6\ u\^2 + 18\ a\^2\ u\^2 - 24\ a\^3\ u\^2 + 8\ a\^4\ u\^2 - 44\ a\^5\ u\^2 + 106\ a\^6\ u\^2 - 64\ a\^7\ u\^2 + 282\ a\^8\ u\^2 - 92\ a\^9\ u\^2 + 228\ a\^10\ u\^2 - 48\ a\^11\ u\^2 + 60\ a\^12\ u\^2 + 4\ a\^14\ u\^2 - 4\ u\^3 + 12\ a\ u\^3 - 32\ a\^2\ u\^3 + 60\ a\^3\ u\^3 - 92\ a\^4\ u\^3 + 120\ a\^5\ u\^3 - 120\ a\^6\ u\^3 + 120\ a\^7\ u\^3 - 60\ a\^8\ u\^3 + 60\ a\^9\ u\^3 + 16\ a\^10\ u\^3 + 12\ a\^11\ u\^3 + 28\ a\^12\ u\^3 + 8\ a\^14\ u\^3 + u\^4 - 6\ a\ u\^4 + 22\ a\^2\ u\^4 - 56\ a\^3\ u\^4 + 114\ a\^4\ u\^4 - 188\ a\^5\ u\^4 + 264\ a\^6\ u\^4 - 312\ a\^7\ u\^4 + 321\ a\^8\ u\^4 - 278\ a\^9\ u\^4 + 210\ a\^10\ u\^4 - 128\ a\^11\ u\^4 + 68\ a\^12\ u\^4 - 24\ a\^13\ u\^4 + 8\ a\^14\ u\^4)\); \)\)], "Input"], Cell[BoxData[ \({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)\)}; \n{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}; \n{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)\)\)\)}; \)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Factor[g\ y\_1\^2 - \((f /. x -> x\_1)\)]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Factor[g\ y\_2\^2 - \((f /. x -> x\_2)\)]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Factor[g\ y\_3\^2 - \((f /. x -> x\_3)\)]\)], "Input"], Cell[BoxData[ \(0\)], "Output"] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell["Theorem 5.3", "Subsection"], Cell[BoxData[ \(\(f = x \((x - b)\) \((x - \(a\^2\) b)\); \)\)], "Input"], Cell[CellGroupData[{ Cell[TextData[{ " ", StyleBox["t", FontSlant->"Italic"], " in terms of ", StyleBox["u", FontSlant->"Italic"] }], "Subsubsection"], Cell[BoxData[ \(\(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; \)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(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] \)], "Input"], Cell[BoxData[ \(\(-4\)\ b\ u\ \((\(-a\) + u - 2\ a\ u + a\^2\ u)\)\ \((1 - a + a\^2 - a\^3 + 4\ a\^2\ u - 3\ a\^3\ u + a\^4\ u)\)\ \((1 + a + 2\ a\ u - 2\ a\^2\ u + 4\ a\ u\^2 - 3\ a\^2\ u\^2 + a\^3\ u\^2)\)\ \((1 + 2\ a\^2 + a\^4 - 2\ a\ u + 4\ a\^2\ u - 4\ a\^3\ u + 4\ a\^4\ u - 2\ a\^5\ u + 4\ a\ u\^2 - 7\ a\^2\ u\^2 + 6\ a\^4\ u\^2 - 4\ a\^5\ u\^2 + a\^6\ u\^2)\)\ \((1 + 2\ a\^2 + a\^4 - 4\ a\ u + 8\ a\^2\ u - 8\ a\^3\ u + 8\ a\^4\ u - 4\ a\^5\ u + 4\ u\^2 - 8\ a\ u\^2 + 14\ a\^2\ u\^2 - 32\ a\^3\ u\^2 + 40\ a\^4\ u\^2 - 24\ a\^5\ u\^2 + 6\ a\^6\ u\^2 + 16\ a\^2\ u\^3 - 60\ a\^3\ u\^3 + 88\ a\^4\ u\^3 - 64\ a\^5\ u\^3 + 24\ a\^6\ u\^3 - 4\ a\^7\ u\^3 + 16\ a\^2\ u\^4 - 56\ a\^3\ u\^4 + 81\ a\^4\ u\^4 - 64\ a\^5\ u\^4 + 30\ a\^6\ u\^4 - 8\ a\^7\ u\^4 + a\^8\ u\^4)\)\)], "Output"] }, Open ]], Cell[BoxData[ \({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}; \n{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\)\)}; \n{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) \)\)\)}; 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