y^2 = a*x^4+b*x^3+c*x^2+d*x+e
is birationally equivalent to an elliptic curve in standard Weierstrass form y^2=cubic(x).
How to I get sage to exhibit/find the birational transformation that accomplishes that?

Warren D Smith warren.wds AT gmail.com

1 | initial version |

y^2 = a*x^4+b*x^3+c*x^2+d*x+e
is birationally equivalent to an elliptic curve in standard Weierstrass form y^2=cubic(x).
How to I get sage to exhibit/find the birational transformation that accomplishes that?

Warren D Smith warren.wds AT gmail.com

2 | No.2 Revision |

`y^2 = `~~a~~*x^4+b*x^3+c*x^2+d*x+e
a*x^4+b*x^3+c*x^2+d*x+e

is birationally equivalent to an elliptic curve in standard Weierstrass form ~~y^2=cubic(x).
~~`y^2=cubic(x)`

.
How to I get sage to exhibit/find the birational transformation that accomplishes that?

Warren D Smith warren.wds AT gmail.com

3 | No.3 Revision |

```
y^2 = a*x^4+b*x^3+c*x^2+d*x+e
```

is birationally equivalent to an elliptic curve in standard Weierstrass form `y^2=cubic(x)`

.
How to I get sage to exhibit/find the birational transformation that accomplishes that?

[I am specifically interested in knowing all about this class of curves:
8*D*y^2 = (x-2)*(x-1)*(x+1)*(x+2)
for integer D. I am pretty new to both SAGE and elliptic curves.]

4 | eq formatting |

```
y^2 = a*x^4+b*x^3+c*x^2+d*x+e
```

is birationally equivalent to an elliptic curve in standard Weierstrass form `y^2=cubic(x)`

.
How to I get sage to exhibit/find the birational transformation that accomplishes that?

[I am specifically interested in knowing all about this class of curves:
~~8~~*D*y^2 `8*D*y^2 `

`= `

for integer D. I am pretty new to both SAGE and elliptic curves.]~~(x-2)~~(x-2)*(x-1)*(x+1)*(x+2)*(x-1)*(x+1)*(x+2)

5 | retagged |

```
y^2 = a*x^4+b*x^3+c*x^2+d*x+e
```

is birationally equivalent to an elliptic curve in standard Weierstrass form `y^2=cubic(x)`

.
How to I get sage to exhibit/find the birational transformation that accomplishes that?

[I am specifically interested in knowing all about this class of curves:
`8*D*y^2 = (x-2)*(x-1)*(x+1)*(x+2)`

for integer D. I am pretty new to both SAGE and elliptic curves.]

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