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Let GF(q) be a finite field over GF(p), p prime. I want to find a primitive element gamma of G(q) and then find the minimal polynomial of gamma^j over GF(p), j an integer.

Is there a default way to do this?

Find the minimal polynomial of an element over a finite field Let GF(q) be a finite field over GF(p), p prime. I want to

Providing some context: i want to create a class to operate on isogeny graphs of elliptic curves. So it should have the $j$-invariants (integers modulo $p$) as nodes and the existence of $l$-isogenies as edges.

To compute the edges i need to do some calculations on GF(p) and some others on the ring PolynomialRing(GF(p), ['X', 'Y']).

How do i make sure the operations happen on their specific rings and don't change the field outside of the class?

So i have a polynomial over 2 variables:

R.<X, Y> = GF(8009)[]
Phi = -X^2*Y^2 + X^3 + 1488*X^2*Y + 1488*X*Y^2 + Y^3 \
    - 162000*X^2 + 40773375*X*Y - 162000*Y^2 \
    + 8748000000*X + 8748000000*Y - 157464000000000

I want to know for what Y values the polynomial has a solution with X = 33. I've tried using the solve method:

Phi(33, Y).roots(Y)

But this yields the error

AttributeError: 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular' object has no attribute 'roots'

I also tried the following:

Phi(33).roots()

It yields the error

TypeError: number of arguments does not match number of variables in parent

So what is the correct way to do this?

Suppose i have declared many varibles and a polynomial using them

x, y, z = var("x y z")
poly =  x^3*y*z + x^2*y^2 + 3*x^3 + 2*x^2 + x*y + x*z + 1

How can i simplify the expression is such a way that it is written as a polynomial over x? I mean something like

(...)*x^3 + (...)*x^2 + (...)*x +...

So i have the following functions:

a,b,c,x,y,p,q,r,s = var("a b c x y p q r s")
g(x,y) = a*x^2 + b*x*y + c*y^2
f(x,y) = f(x,y) = g(p*x + q*y, r*x + s*y)
#p * s - q * r = 1

I calculated the discriminant of the binary form f

valueA, valueC = f(1,0),f(0,1)
valueB = f(1,1) - valueA - valueC
D = valueB^2 - 4 * valueA * valueC

I obtain a lengthy expression:

D =  b^2*q^2*r^2 - 4*a*c*q^2*r^2 - 2*b^2*p*q*r*s + 8*a*c*p*q*r*s + b^2*p^2*s^2 - 4*a*c*p^2*s^2

If i apply the relation p * s - q * r = 1 i should obtain D = ( p * s - q * r )^2 * (b^2 - 4 * a *c). The thing is, i don't know how to apply it in a way that makes it simplify like that.

I tried calling simplify and full_simplify on D/(p * s - q * r - 1) but it didn't simplify this neatly. Also using substitute ({p : (1 + q * r) /s}) isn't enough.

I'm trying to get an elliptic curve plot, but the points are too thick and the resolution too low. This makes the points stick together in a mess. The following code

E = EllipticCurve(GF(next_prime(20000)),[0,1])
E.plot()

results in this image(https:// imgur.com/a/aMFpFXN). How can i make the points less thick/the resolution bigger?

Thanks, it works.

I was trying to use a lagrange multiplier to find the local maxima of a simple function over the circle:

x,y,c = var('x y c')

f(x,y) = x**2 + y**2 - 1
g(x,y) = x+y
F = diff(f)
G = diff(g)

polys = [f]
for i in range(len(F)):
    p = F[i] - c * G[i]
    polys.append(p)

list_solutions = solve(polys,x,y,c)

for solution in list_solutions:
    print solution
    print f({x : solution[0] , y : solution[1]})
    print '\n'

At 'print solution' i get the first extremum:

[x == 1/2*sqrt(2), y == 1/2*sqrt(2), c == sqrt(2)]

But at the 'print f({x : solution[0] , y : solution[1]})' line the following error occurs:

TypeError: no canonical coercion from <type 'dict'> to Callable function ring with arguments (x, y)

So i guess x == 1/2*sqrt(2) and the like can't be seem as numbers. I would like to be able to evaluate f(x,y) at the point, so what can i do? Also, the lagrange multiplier algorithm must be on Sage already, but Google gives me nothing. If you can show its command that would be a plus.

I was trying to represent S³ as a three dimensional manifold, with coordinates (x,y,z,w) in R⁴, and make the transition map from the upper cap w > 0 to the lateral cap z<0, with the charts being the graphs of the caps as functions. I came up with the following code:

M = Manifold(3, 'S^3')

N = M.open_subset('N') 
projN.<x,y,z> = N.chart() 

E = M.open_subset('E') 
projE.<x,y,w> = E.chart()

ProjNE = projN.transition_map(projE,
                [x,y, sqrt(1-x^2-y^2-z^2)], intersection_name='D',
                restrictions1= z < 0, restrictions2= w>0)

It sounds reasonable, but calling

ProjNE.inverse()

failed. No problem, i tried using

ProjNE.set_inverse(x,y, -sqrt(1-x^2-y^2-w^2))

but i got the following warning:

Check of the inverse coordinate transformation:
  x == x  *passed*
  y == y  *passed*
  z == -abs(z)  **failed**
  x == x  *passed*
  y == y  *passed*
  w == abs(w)  **failed**
NB: a failed report can reflect a mere lack of simplification.

i don't know why the test is failing. The math sounds ok, where did it go wrong?