## Calculate the discriminant and the conductor of an elliptical curve using magma

To calculate the minimum discrimination and the conductor of an elliptical curve using magma for example this elliptic curve, we use this command

E: = EllipticCurve ((0,8,0,48,0));

E;

Elliptical curve defined by y ^ 2 = x ^ 3 + 8 * x ^ 2 + 48 * x on the rational field

F: = MinimalModel (E);

F;

Elliptical curve defined by y ^ 2 = x ^ 3 – x ^ 2 + 2 * x – 2 on the rational field

D: = Discriminant (F);

N: = conductor (E);

My quastion is how to calculate the discriminant and the conductor when the curve has a variable coefficient for example this elliptical curve

$$y ^ 2 = x (x-a) (x-D ^ {p} zeta ^ {k})$$
Or $$a, D$$ are integers and $$zeta ^ {k}$$ is the k-th power of unity

## Is there a code in MAGMA to check the absolute irreducibility of a polynomial on finite fields?

I found a code on the magma k: = FieldOfGeometricIrreducibility (C); Irreducible components (BaseChange (C, k)); I just want to know if this command checks the irreducibility or the absolute irreducibility of the polynomial. Help me nicely with this look.

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## finite fields – Construction of \$ mbox {GRS} \$ extended code in Magma Software

Let $${ bf a} = [a_1, . . . , a_n]$$ to be a sequence of $$n$$ distinct elements taken in the finished body $$mathbb {F} _q$$ and let $${ bf v} = [v_1, . . . , v_n]$$ to be a sequence of n non-zero elements of $$mathbb {F} _q$$. Let $$k$$ to be a non-negative integer. Given that $$bf a$$, $$bf v$$ , and $$k$$ in magma the order $$mbox {GRS} _k ({ bf a, v})$$ builds the generalized code of Reed – Solomon on $$mathbb {F} _q$$. It's a $$[n, k’]$$ code as $$k leq$$.

In addition, we consider the extension $$mbox {GRS}$$ noted code $$mbox {GRS} _k ({ bf a, v}, infty)$$. The code is given by

$$mbox {GRS} _k ({ bf a, v}, infty) = {(v_1f (a_1), …, v_nf (a_n), f_ {k – 1}): quad f (x) in mathbb {F} _q[x], quad deg (f) ≤ k – 1 },$$

or $$f_ {k – 1}$$ represents the coefficient of $$x ^ {k – 1}$$ in $$f (x)$$. It is known that the extension of the GRS code is still an MDS code. So,
$$mbox {GRS} _k ({ bf a, v}, infty)$$ is a $$q-$$ary $$[n + 1, k, n − k + 2]$$ $$mbox {MDS}$$ code.

My question:

Suppose we built the code $$mbox {GRS} _k ({ bf a, v})$$ in magma. How to obtain $$mbox {GRS} _k ({ bf a, v}, infty)$$ of $$mbox {GRS} _k ({ bf a, v})$$ in the Magma software?

My essay:

First I build the code $$C = mbox {GRS} _k ({ bf a, v})$$ in magma. Then by
by applying the command B = ExtendCode (C), I build an extension of C. But this command does not produce the code $$mbox {GRS} _k ({ bf a, v}, infty)$$since the B code must be an MDS code, but in some examples, the ExtendCode (C) command generates a non-MDS code.

More precisely, suppose that two vectors $${ bf a} = [a_1, . . . , a_n]$$ and $${ bf v} = [v_1, . . . , v_n]$$ we built the $$mbox {GRS} _k ({ bf a, v})$$. Now, how to define two vectors $${ bf a} = [a’_1, . . . , a’_{n+1}]$$ and $${ bf v} & # 39; = [v’_1, . . . , v’_{n+1}]$$ of $$bf a$$ and $$bf v$$ such as
$$mbox {GRS} _k ({ bf a,, v ))$$ bean $$[n+1,k]$$ MDS code.

Thank you for all suggestions.