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2015-03-02 05:38:57 -0500 | commented question | Constructing subgroups by intersection Okay, thanks for your response! What I really want to look at are the generators for such subgroups.. |

2015-02-28 10:25:57 -0500 | commented question | Constructing subgroups by intersection Sorry, I should have put curly brackets around $M\left(N\right)$, so it's just a matrix. Specifically, I want to construct: $$G_0\left(N\right) = { \left( \begin{array}{cccc} \mathbb{Z} & \mathbb{Z} & \mathbb{Z} & N\mathbb{Z} \ N\mathbb{Z} & \mathbb{Z} & N\mathbb{Z} & N^2 \mathbb{Z} \ \mathbb{Z} & \mathbb{Z} & \mathbb{Z} & N \mathbb{Z} \ \mathbb{Z} & \mathbb{Z} & \mathbb{Z} & \mathbb{Z} \end{array} \right) } \cap {Sp}\left(4,\mathbb{Z}\right)$$ We can define congruence subgroups of the modular group in this way, but I want to do the same thing for subgroups of $Sp\left(4,\mathbb{Z}\right)$. Thanks for your help! |

2015-02-27 12:03:03 -0500 | asked a question | Constructing subgroups by intersection I'd like to construct a subgroup of $Sp\left(4,\mathbb{Z}\right)$ of the form: $$G_0\left(N\right) = M\left(N\right) \cap {Sp}\left(4,\mathbb{Z}\right)$$ where $M\left(N\right)$ is a $4\times4$ matrix over the integer ring with elements that are multiples of the integer $N$. I think I know how to construct such an $M\left(N\right)$ for a given $N$, but how does one then construct such a subgroup $G_0\left(N\right)$? Thanks! |

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2013-01-05 04:24:24 -0500 | asked a question | Check whether a modular subgroup is congruence from its generators I want to input the generators of a specific subgroup of the modular group $\Gamma = PSL(2,\mathbb{Z})$ as explicit $2\times 2$ matrices, and have Sage tell me whether that subgroup is congruence. Clearly I need to use the command |

2012-09-01 01:36:43 -0500 | asked a question | Plotting fundamental domains of modular subgroups from generators and cusp widths I have a number of genus zero, index 24 modular subgroups (some congruence, some not). For each of these subgroups, I have the cusp widths, and a set of generators as explicit 2x2 matrices. How can I use these to plot the fundamental domain for each group? I suspect a lot of this material is in the Thanks! |

2012-08-06 02:50:23 -0500 | marked best answer | Permutation Representations and the Modular Group Hi, Given a congruence subgroup, you can get another data structure for it given exactly by the action of the standard generators (L = parabolic fixing infinity, R = parbolic fixing 0, S2 = element of order 2, S3 = element of order 3). You may proceed as follows for the first example of the list Warning: it is the right action on right coset (ie of the form Hg). Note that you can also plot a fundamental domain using Farey symbols (Kulkarni method) Vincent |

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2012-08-06 02:49:52 -0500 | commented answer | Permutation Representations and the Modular Group Great! Thanks very much |

2012-08-05 22:34:42 -0500 | commented answer | Permutation Representations and the Modular Group Thanks very much! And I have a follow-up question too: Is there an easy way to find the permutation representations of PSL(2,Z) (rather than SL(2,Z)) on the cosets of these congruence subgroups? I notice there is a projective_index() command, so is there anything similar which could do the job here? Many thanks, |

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2012-08-03 04:19:42 -0500 | asked a question | Permutation Representations and the Modular Group Hi! Given a congruence subgroup of the modular group G = SL2Z, how can one use Sage to find the permutation representations of G on the cosets of each of those normal subgroups? Specifically I am looking at the subgroups on page 22 here: http://arxiv.org/pdf/1201.3633v2.pdf Many thanks! |

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