Content deleted Content added
m OEIS in section external links: use dedicated {{OEIS el}}. using the long formal name (via WP:JWB) |
Ira Leviton (talk | contribs) m Fixed a typo found with Wikipedia:Typo_Team/moss. |
||
Line 11:
== Geometry of the modular curve ==
[[Image:Modknot11.png|thumb|Knot at infinity of {{math|''X''<sub>0</sub>(11)}}]]
The classical modular curve, which we will call {{math|''X''<sub>0</sub>(''n'')}}, is of degree greater than or equal to {{math|2''n''}} when {{math|''n'' > 1}}, with equality if and only if {{mvar|n}} is a prime. The polynomial {{math|Φ<sub>''n''</sub>}} has integer coefficients, and hence is defined over every field. However, the coefficients are sufficiently large that computational work with the curve can be difficult. As a polynomial in {{mvar|x}} with coefficients in {{math|'''Z'''[''y'']}}, it has degree {{math|''ψ''(''n'')}}, where {{mvar|ψ}} is the [[Dedekind psi function]]. Since {{math|Φ<sub>''n''</sub>(''x'', ''y'') {{=}} Φ<sub>''n''</sub>(''y'', ''x'')}}, {{math|''X''<sub>0</sub>(''n'')}} is symmetrical around the line {{math|''y'' {{=}} ''x''}}, and has singular points at the repeated roots of the classical modular polynomial, where it crosses itself in the complex plane. These are not the only singularities, and in particular when {{math|''n'' > 2}}, there are two
== Parametrization of the modular curve ==
| |||