## 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

## machine learning – Linear Discriminant Analysis (LDA) before or after cross-validation k-fold

I have functionality extracted from a small dataset, I would like to reduce the dimensions using LDA. You also want to perform an SVM classification with cross-validation k-fold.
My question is:
What would be the best practice: do the LDA before the CV, or do the LDA within the CV (i.e. for each train and test fold)?

## algebraic number theory – Question on the lemma involving a field extension discriminant

I am trying to understand a proof given in Neukirch's algebraic number theory of the following lemma: Let $$alpha_1, …, alpha_n$$ to be a base of $$L / K$$ which is contained in $$B$$, the algebraic closure of $$A$$ in $$L$$, discriminating $$d = d ( alpha_1, …, alpha_n)$$. Then we have $$dB subseteq A alpha_1 + cdots A alpha_n$$. The proof is as follows: if $$alpha = a_1 alpha_1 + cdots a_n alpha_n in B$$, $$a_j in K$$, then the $$a_j$$ are solutions of the system of linear equations $$Tr_ {L / K} ( alpha_i alpha) = sum_jTr_ {L / K} ( alpha_i alpha_j) a_j$$, and like $$Tr_ {L / K} ( alpha_i alpha) in A$$, they are given as the quotient of an element of $$A$$ by the determinant $$det (Tr_ {L / K} ( alpha_i alpha)) = d$$.

My question is, why the $$a_j$$ be expressed as such a quotient? I don't see how the matrix equation from the system of linear equations above implies this.

Is it because we can $$d$$ of a different row of $$j$$ and divide by $$d$$?

## Quadratic number field with discriminant -67

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Let $$K = mathbb Q ( sqrt 5)$$ and $$varepsilon = 9 + 4 sqrt 5$$ its fundamental unit. For all $$n geq 3$$, let $$L_n = K ( zeta_ {5 times 2 ^ n}, sqrt[5times 2^n]{ varepsilon})$$, or $$zeta_ {5 times 2 ^ n}$$ is a primitive $$5 times 2 ^ n$$-th root of unity. In other words, $$L_n$$ is the dividing field of $$X ^ {5 times 2 ^ n} – varepsilon$$ more than $$K$$.

I would like to calculate, or at least the upper limit, the absolute discriminant of $$L_n$$. Thank you in advance.

## Relationship between discriminant and coefficients of a quadratic function

To answer question (b), the problem requires that I use the discriminant. However, I still do not understand how I would realize that I should make use of it. Can any one please explain the relationship between the two inequalities?

P.S.
For those who think that I am looking for answers to a duty, I have the answer to (b), so it is not necessary to provide it.

## Understand the $p$ part of the discriminant of a totally real number field with a prime number greater than $p$

Let $$K$$ to be a totally real number field of Galois, and suppose that there is only one prime number above $$p$$, with branching index $$leq p-1$$. Yes $$K_p$$ is the completion of $$K$$ at the highest $$p$$, the claim is that the $$p$$-part of the discriminant of $$K$$ is equal to the discriminant of $$K_p$$.

I came across this information by reading Washington's "Introduction to Cyclotomic Fields," where he mentions in the proof of proposition 5.33 that "the $$p$$-part of the discriminant of $$K$$ is equal to the discriminant of $$K_p$$& # 39 ;, where the configuration is as described above. It's not clear to me how the basics for $$mathcal O_K$$ and $$mathcal O_ {K_p}$$ are related, so I'm not sure how to make sense of it. I've tried to decompress it a little bit by looking at the example where $$K = mathbb Q ( zeta_p) ^ +$$ is the maximum real subfield of $$mathbb Q ( zeta_p)$$, in which case $$(p) = (1- zeta_p) ^ {p-1}$$ is totally branched in $$mathbb Q ( zeta_p)$$, so $$K$$ satisfies the hypotheses. But even in this example, I have trouble calculating the relevant discriminants, let alone understanding this in general …

## polynomials – Find $text {k}$ such as $:$ $text {discriminant} =$ 0

I tried to find $$text {k}$$ such as $$:$$
$$text {discriminant} left [ text{discriminant}left [ z^{,2}left ( z^{,2}- 1 right )^{,2}left ( z^{,2}+ 1 right )+ left ( y^{,2}- z^{,2} right )left ( y^{,4}+ z^{,6}- z^{,4}- z^{,2} right )- text{k}left ( z^{,4}- 1 right )z^{,2}left ( y^{,2}- z^{,2} right ),,y right ] ,, z right]= 0$$
or $$:$$
$$text {discriminant} left [ text{discriminant}left [ z^{,2}left ( z^{,2}- 1 right )^{,2}left ( z^{,2}+ 1 right )+ left ( y^{,2}- z^{,2} right )left ( y^{,4}+ z^{,6}- z^{,4}- z^{,2} right )- text{k}left ( z^{,4}- 1 right )z^{,2}left ( y^{,2}- z^{,2} right ),,z right ] ,, y right]= 0$$
Then I used Wolfram Alpha but it can$$&$$t bring a polymonial of $$text {k}$$ $$,$$ see $$:$$

https://www.wolframalpha.com/input/?i=discriminant%5By%5E6%2By%5E2z%5E6%2Bz%5E2-y%5E2z%5E2 (1% 2By% 5E2% 2Bz% 5E2) -k ( 1% 2Bz% 5E2) z% 5E2 (y% 2Bz) (yz) (z-1) (z% 2B1)% 5D, z% 5D

$$text {discriminant} left [ z^{,2}left ( z^{,2}- 1 right )^{,2}left ( z^{,2}+ 1 right )+ left ( y^{,2}- z^{,2} right )left ( y^{,4}+ z^{,6}- z^{,4}- z^{,2} right )- text{k}left ( z^{,4}- 1 right )z^{,2}left ( y^{,2}- z^{,2} right ),,z right ]= 0$$

https://www.wolframalpha.com/input/?i=discriminant%5By%5E6%2By%5E2z%5E6%2Bz%5E2-y%5E2z%5E2 (1% 2By% 5E2% 2Bz% 5E2) -k ( 1% 2Bz% 5E2) z% 5E2 (y% 2Bz) (yz) (z-1) (z% 2B1)% 5D, y% 5D

$$text {discriminant} left [ z^{,2}left ( z^{,2}- 1 right )^{,2}left ( z^{,2}+ 1 right )+ left ( y^{,2}- z^{,2} right )left ( y^{,4}+ z^{,6}- z^{,4}- z^{,2} right )- text{k}left ( z^{,4}- 1 right )z^{,2}left ( y^{,2}- z^{,2} right ),,y right ]= 0$$

But $$:$$

https://www.wolframalpha.com/input/?i=discriminant%5Bdiscriminant%5By%5E6%2By%5E6%2By%5E6%2Bz%5EB2%2%%%%%%%%%%>. k (1% 2Bz% 5E2) z% 5E2 (y% 2Bz) (yz) (z-1) (z% 2B1)% 5D, y% 5D, z% 5D

I've used these to solve an inequality $$($$ his original idea is a IMO problem $$)$$ $$.$$ How can I get this $$text {k}$$ $$?$$ I need help $$!$$

Good luck to everyone $$!$$