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2020-08-24 23:01:56 +0200 | marked best answer | Factoring a polynomial over a finite Field Hello I'm following a 101 algebra course, and for example, I would like to factor a polynomial on a finite field like F_9 (F_9 == ZZ/9ZZ is a field because 9 is a power of a prime number, 3) i get an error message i'm running sage 6.3 notebook on windows through Oracle VM winbox. I'm a totally new user, and i've looked at the tutorial and the forum but couldn't find any example or reason why this would not work. thank you for your help. |

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2020-06-26 05:00:00 +0200 | asked a question | SageMath: defining class of functions on Elliptic Curves In SageMath, I would like to manipulate rational functions on elliptic curves (defined on finite fields). For example, for $P = (x,y)$ on some curve $E$ $$f = x+y-12$$ $$g = \frac{x+y-3}{(x-3)^2} $$ etc. Is there a natural class? I am looking to make a toy example with pairings, so I need to define stuff like $$P \rightarrow f_P$$ where $$f_P:Q \rightarrow f_P(Q)$$ is a function I can't see how to do that, and I'm able to make computations if I define but I can only compute the values taken by the function $f_P$, I cannot "see" the function $f_P$. Basically I'm trying as an exercise to re-write the following Magma code to SageMath: http://www.craigcostello.com.au/pairi... EDIT My question hasn't attracted much interest so let me give a more concrete example. Define the following: I would have hoped for Instead it raises the following error: |

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2017-03-08 09:32:07 +0200 | commented answer | n-torsion subgroups on Elliptic Curves defined on some field thank you very much, i need some time to digest it all, because i'm a beginner/amateur in this topic. i don't really understand what f and g are. |

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2017-02-10 09:52:38 +0200 | commented answer | Elliptic Curves defined over Z/nZ rings for general n i think they should be well understood to some extent. there are the basis of some powerful popular methods for integer factorization or proving that a large number is prime for example. |

2017-02-10 02:12:10 +0200 | commented question | Elliptic Curves defined over Z/nZ rings for general n do you see a way around that ? |

2017-02-09 16:25:57 +0200 | asked a question | Elliptic Curves defined over Z/nZ rings for general n An Elliptic curve is the union of its affine part Eaff(Z/nZ)={ [x,y,1] in P2(Z/nZ) such that y2 =x3+ax+b } and the point at infinity O = [0,1,0] P2(Z/nZ) is the projective plane: T = { (x,y,z) in (Z/nZ)^3 such that gcd(x,y,z,n) = 1} P2(Z/nZ) = (T / ~) where ~ is the equivalence relation defined by (x,y,z) ~ (x0,y0,z0) iff there is an INVERSIBLE element u in Z/nZ ⇤ such that (x,y,z) = u(x0,y0,z0). I found those doc pages http://fe.math.kobe-u.ac.jp/icms2010-... http://doc.sagemath.org/html/en/refer... I think to define an Elliptic Curve here with
but is not working. is there a method to get all the points of this Elliptic Curve ? its cardinality etc ???? |

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2017-02-08 15:26:56 +0200 | asked a question | n-torsion subgroups on Elliptic Curves defined on some field if E an elliptic is defined on K, then E[n] is a subgroup of an Elliptic curve defined on an algebraic closure of K, its the set of points P of this curve whose order is dividing n G = E.torsion_subgroup() gives the torsion subgroup in E(K), but is there a method giving the E[n] ? For example, for the Elliptic Curve y^2=x^3-2 defined on Q if we denote by alpha the cubic root of 2 and J a complex cubic root of unity, and O the point at infinity E[2] = {O,(alpha,0), (alpha * J,0),(alpha * J^2,0)} I tried the following
and got [pretty close but not the right result, and not in an algebraic form anyway] thanks |

2016-12-03 17:58:15 +0200 | commented answer | Action of a permutation on a list (and other objects) @tmonteil could you have a second look please ? thanks |

2016-12-01 11:08:52 +0200 | edited question | Action of a permutation on a list (and other objects) Hi, i'm a beginner user in Sage and I am currently trying to do a few exercises about the Rubik's Cube from the book "Adventures in Group Theory" from D. Joyner from a general move on the rubik cube (defined as a permutation of its 48 facets) i want to define its restriction to the subset of corner (or edge) facets.
basically i would like to look at the action of this permutation on those lists, and i should have for the corners (1,40,41,17)(3,38,43,19)( 6,35,46,22 )(8,33,48,24)(9,14,16,11)(25,27,32,30) and for the edges (4,37,44,20)(5,36,45,21)(10,12,15,13)(26,29,31,28) i see there is a method called _r_action but it doesn't look like its working here
I guess another way of doing it would be to define a (surjective) homomorphism from S(48) into S(edges) and S(corners) but i don't know how to do it EDIT: i am trying a solution along these lines by trying to define an homomorphism. lets try with a simpler example
I think i tried the same syntax than here http://doc.sagemath.org/html/en/refer... surely i must have done something wrong , but what ? thanks |

2016-11-29 09:43:02 +0200 | marked best answer | using GAP: Group Theory I'm trying this code base on the book "Adventures in Group Theory" from D.Joyner
However, there seems to be a signature problem the following commands don't work: (the first one was from the book, I tried myself the second one) Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_92.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# - File "/private/var/folders/gm/z065gk616xg6g0xgn4c7_bvc0000gn/T/tmpHRYMWK/___code___.py", line 2, in <module> exec compile(u'G2=PermutationGroup(gens) File "", line 1, in <module> File "/Applications/SageMath/local/lib/python2.7/site-packages/sage/groups/perm_gps/permgroup.py", line 343, in PermutationGroup raise TypeError("gens must be a tuple, list, or GapElement") TypeError: gens must be a tuple, list, or GapElement Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "_sage_input_79.py", line 10, in <module>
exec compile(u'open("___code___.py","w").write("# - File "/private/var/folders/gm/z065gk616xg6g0xgn4c7_bvc0000gn/T/tmpOxJAnr/___code___.py", line 2, in <module> exec compile(u'G2=PermutationGroup(list(gens)) File "", line 1, in <module> File "/Applications/SageMath/local/lib/python2.7/site-packages/sage/groups/perm_gps/permgroup.py", line 345, in PermutationGroup canonicalize=canonicalize, category=category) File "/Applications/SageMath/local/lib/python2.7/site-packages/sage/groups/perm_gps/permgroup.py", line 419, in __init__ for cycle in x: TypeError: 'sage.libs.gap.element.GapElement_Permutation' object is not iterable I cannot make much of the documentation. Thanks. |

2016-11-27 12:03:15 +0200 | commented answer | Action of a permutation on a list (and other objects) thank you , but you are not really answering to my question. There are 24 edge facets and 24 corner facets, and i want to see the 2 natural projections of p on S(24) _r_action doesn't work on P |

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