1 | initial version |

Since computation of `f2`

involves division by `S((y + 4)^2)`

, you'd need to define `S`

as

```
S = K.quotient(y^2 - x^3 - 4*x).fraction_field()
```

2 | No.2 Revision |

Since computation of `f2`

involves division by `S((y + 4)^2)`

, you'd need to define `S`

as

```
S = K.quotient(y^2 - x^3 - 4*x).fraction_field()
```

to get `S(y - 2*x)`

in `I.

3 | No.3 Revision |

Since computation of `f2`

involves division by `S((y + 4)^2)`

, you'd need to define `S`

as

```
S = K.quotient(y^2 - x^3 - 4*x).fraction_field()
```

to get `S(y - 2*x)`

in ~~`I.~~`I`

.

4 | No.4 Revision |

~~Since ~~First, multivariate ideal machinery in Sage is known to be buggy for non-fields and `Integers(11)`

is not recognized as a field. Replace it with `GF(11)`

.

Second, since computation of `f2`

involves division by `S((y + 4)^2)`

, ~~you'd ~~you indeed need to define `S`

~~as~~as a fraction field:

```
S = K.quotient(y^2 - x^3 - 4*x).fraction_field()
```

to get `S(y - 2*x)`

in `I`

.

5 | No.5 Revision |

First, the multivariate ideal machinery in Sage is known to be buggy ~~for non-fields ~~over non-fields, and `Integers(11)`

is not recognized as a field. Replace it with `GF(11)`

.

Second, since computation of `f2`

involves division by `S((y + 4)^2)`

, you indeed need to define `S`

as a fraction field:

```
K.<x, y> = GF(11)[]
S = K.quotient(y^2 - x^3 - 4*x).fraction_field()
```

to get `S(y - 2*x)`

in `I`

.

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