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2016-08-10 16:48:49 -0500 answered a question How do I compute modular polynomials?

I think this might be what he was looking for:

http://combinat.sagemath.org/doc/refe...

In particular, look at sage.modular.ssmod.ssmod.Phi_polys(L, x, j)

2016-03-04 21:01:46 -0500 asked a question Elliptic curve isogenies: kernel polynomial not monic

Hello, I was trying to follow the example here to create a kernel equal to the full 2 torsion. But I ran into the problem of not having a monic kernel polynomial. This is what I have

sage: p = 22031
sage: K = GF(p)
sage: F.<z> = K[]
sage: L = GF(p^2,'a',modulus=z^2+2); a = L.gen()
sage: E = EllipticCurve(K,[6486*a+8098, 12871*a+17004])
sage: ker_list = E_B.division_polynomial(2).list()
sage: phi = EllipticCurveIsogeny(E, ker_list); phi
ValueError: The kernel polynomial must be monic.

I've tried to salvage the situation by feeding the list of 2-torsion points into the function, but I do not get the same isogeny.

sage: E = EllipticCurve(GF(3), [0,0,0,1,1])
sage: ker_list = E.division_polynomial(2).list()
sage: phi = EllipticCurveIsogeny(E, ker_list)
sage: print phi
Isogeny of degree 4 from Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 3 to Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 3
sage: ETwoTors = E(0).division_points(2)
sage: phi = EllipticCurveIsogeny(E, ETwoTors)
sage: print phi
Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 3 to Elliptic Curve defined by y^2 = x^3 + 2*x over Finite Field of size 3

What I am really after is to let $\phi$ be the multiplication by 2 endomorphism.

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2014-11-03 20:24:20 -0500 asked a question Virtualbox and Chromium

Hi, I've been using sage.5.11 on VirtualBox running on a Windows 7 computer. I would like to know a couple of things:

  1. How to go from the Notebook mode to a command line mode
  2. How to return to the Notebook mode after I clicked on a link to view truncated output
  3. Booting into Command Line SAGE instead of Notebook mode as it is doing currently

Thanks for your help in advance.

2014-08-25 19:28:33 -0500 commented answer Polynomial in $GF(p^2)$

Thank you so much. SAGE would not be as good as it is without people like you.

2014-08-25 03:03:34 -0500 asked a question Polynomial in $GF(p^2)$

Hello, I have a finite field $K=GF(p^2)$ and the polynomial ring $R=K[x,y]$. They are defined as such

p = 11;
K = GF(p^2,'t');
K.inject_variables();
R.<e1,e2> = K[];
f = (-4*t + 3)*e1*e2 + (2*t + 4)*e2^2 + (2*t - 4)*e1 + (-5*t + 3)

I would like to know if it is possible to group/order all the terms with respect to t instead of e1 and e2. That is, I would like to see

g = -4*e1*e2 + 2*e2^2 + 2*e1 - 5;
h = 3*e1*e2 + 4e2^2 + 4*e1 + 3;

such that

g*t + h == f

Thanks for your help!

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2014-06-29 23:57:35 -0500 commented answer Implementing PARI/GP scripts

Thank you so much!

2014-06-29 20:01:58 -0500 asked a question Implementing PARI/GP scripts

Hello, I have the following gp script which I would like to use in SAGE.

Script: http://pages.cs.wisc.edu/~yeoh/nt/sat...
Description: http://pari.math.u-bordeaux.fr/archiv...

This script is used to find cardinality of an elliptic curve over binary fields. I've also taken a look at

http://trac.sagemath.org/ticket/11548

but was unable to implement either.

I thought that just copying and pasting the codes would enable me to call the functions used, but I was wrong. I'm using SAGE on VirtualBox on Windows. Any help will be appreciated!

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2014-06-28 20:15:59 -0500 marked best answer Substitute and evaluate

Hello, I would like to ask a really easy question:

How can I substitute a variable into an equation and have it evaluated simultaneously? If I have $f = a + b$ and would it to return $5$ if $a=2$ and $b=3$, how shall I proceed?

I've tried the following

a, b = var('a, b');
f = a + b;
a = 2; b = 3; f

But it returned a+b, which is not what I had in mind. Thanks for the help in advance!

2014-06-28 20:15:59 -0500 marked best answer Binary fields

Hello, I would like to perform the following on a binary field, i.e. GF(2^m).

  1. Define a polynomial and solve the polynomial over a binary field.
  2. Convert an element of the binary field into a bit string.

For the first, I've tried the following:

K = GF(2^7,'a');
PK.<x>=K[]; #I've also tried "x = PolynomialRing(GF(2^7,'a'),'x').gen"
f = (a^6 + a^3 + a)*x^2 + (a^6 + a^4 + a^3)*x + (a^5 + a^4 + a^3 + a^2 + 1);
print f.roots();

But the error is TypeError: unable to coerce from a finite field other than the prime subfield.

For the second, I would like to know how finite field elements are stored in SAGE, are they stored as vectors?

If you have any resources that could point me in the right direction, I'll be very thankful for your help!

2014-06-28 20:15:46 -0500 marked best answer Evaluating values of the Weierstrass $\wp$-function

I would like to know how can we evaluate the Weierstrass $\wp$-functions. That is, I would like to find $\wp(\theta,\omega,i\omega)$ for some $\theta,\omega\in\mathbb{R}$.

I'm only able to find a function which outputs the Laurent series of the Weierstrass $\wp$-function when an elliptic curve has been entered. Should I evaluate that laurent series?

2014-06-28 20:15:46 -0500 marked best answer Coefficients of infinite polynomial products

I would first like to ask if there are commands for computing infinite products,

$$eg.\quad g=q \prod_{n=1}^{\infty} (1-q^{8n})(1-q^{16n}).$$

If not, are there commands for finite products then? How do you compute $$\quad g=q \prod_{n=1}^{N} (1-q^{8n})(1-q^{16n})?$$

Basically, I'm only interested in the coefficients of $g\theta_2$ and $g\theta_4$ where $g$ is as above and $\theta_t=\sum^{\infty}_{-\infty}q^{tn^2}$.

This link to OEIS seems to have a code for generating $g\theta_2$ in PARI, I always thought that SAGE is able to call pari, but I'm unable to perform the command on SAGE.

I'll appreciate it if anyone can advise me how best to do this or direct me to any resources, thanks!

2014-06-28 20:15:45 -0500 marked best answer Tuples or for loops

Hi everybody, I would like to compute the $L$-function of an elliptic curve of the form $$y^2=x^3-d^2x$$ for multiple values of $d$ at $s=1$. I was hoping to use an $n$-tuple

d=(1,3,5,7,...,n)

with the hope that I could then do

E=EllipticCurve([0,0,0,-(d^2),0])
L = E.lseries().dokchitser()
L(1)

such that at every stage the data would be in a vector/$n$-tuple form, but this is not possible. Does anyone know how I can make this method work? With a for-loop perhaps? Thanks in advance for your help!

2014-06-28 20:15:39 -0500 marked best answer Pari source code

I'm terribly sorry for asking such a basis question, but can anyone tell me I may be able to find an online access to PARI source code similar to the one we have for sage?

http://hg.sagemath.org/sage-main/src/...

If there isn't one, is there an easy way to view the source code?

2014-06-28 20:12:47 -0500 marked best answer Elliptic Curve Rational Field Height CODE

Hi, I would like to know what does the function R() in lines 5060,5061 etc do. This is the code that I'm looking at:

http://hg.sagemath.org/sage-main/file...

Thank you!

2014-06-08 16:33:14 -0500 commented answer Binary fields

P.S. Sorry I can't accept both answers.

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2014-06-08 16:32:14 -0500 marked best answer Binary fields

There is a problem since the variable a you are using in your third line seems to come from a previous computation.

At least on the last version of Sage, if you type

sage: K = GF(2^7,'a');

The python variable a does not point to the generator of K whose name is 'a'

sage: a
NameError: name 'a' is not defined

For this, you have to do:

sage: K.inject_variables()
Defining a

Then, everything seems to work:

sage: PK.<x>=K[];
sage: f = (a^6 + a^3 + a)*x^2 + (a^6 + a^4 + a^3)*x + (a^5 + a^4 + a^3 + a^2 + 1);
sage: print f.roots();
[(a^3 + a, 1), (a^5 + a^3 + a^2 + a, 1)]
2014-06-08 14:22:53 -0500 commented answer Binary fields

This is exactly what I have in mind. Thank you so much. The reverse procedure (if anyone is interested) can be done via `K._cache.fetch_int(0x426)`