Editing Modular curve (section)

== Genus ==
The covering ''X''(''N'') → ''X''(1) is Galois, with Galois group SL(2, ''N'')/{1, −1}, which is equal to PSL(2,&nbsp;''N'') if ''N'' is prime. Applying the [[Riemann–Hurwitz formula]] and [[Gauss–Bonnet theorem]], one can calculate the genus of ''X''(''N''). For a [[prime number|prime]] level ''p'' ≥ 5,

:<math>-\pi\chi(X(p)) = |G|\cdot D,</math>

where χ = 2 − 2''g'' is the [[Euler characteristic]], |''G''| = (''p''+1)''p''(''p''−1)/2 is the order of the group PSL(2, ''p''), and ''D'' = π − π/2 − π/3 − π/''p'' is the [[defect (geometry)|angular defect]] of the spherical (2,3,''p'') triangle. This results in a formula

:<math>g = \tfrac{1}{24}(p+2)(p-3)(p-5).</math>

Thus ''X''(5) has genus 0, ''X''(7) has genus 3, and ''X''(11) has genus 26. For ''p'' = 2 or 3, one must additionally take into account the ramification, that is, the presence of order ''p'' elements in PSL(2, '''Z'''), and the fact that PSL(2, 2) has order 6, rather than 3. There is a more complicated formula for the genus of the modular curve ''X''(''N'') of any level ''N'' that involves divisors of ''N''.

===Genus zero===
In general a '''modular function field''' is a [[Function field of an algebraic variety|function field]] of a modular curve (or, occasionally, of some other [[moduli space]] that turns out to be an [[irreducible variety]]).  [[Genus (mathematics)|Genus]] zero means such a function field has a single [[transcendental function]] as generator: for example the [[J-invariant|j-function]] generates the function field of ''X''(1) = PSL(2, '''Z''')\'''H'''*.  The traditional name for such a generator, which is unique up to a [[Möbius transformation]] and can be appropriately normalized, is a '''Hauptmodul''' ('''main''' or '''principal modular function''', plural '''Hauptmoduln''').

The spaces ''X''<sub>1</sub>(''n'') have genus zero for ''n''&nbsp;=&nbsp;1, ..., 10 and ''n'' = 12.  Since each of these curves is defined over '''Q''' and has a '''Q'''-rational point, it follows that there are infinitely many rational points on each such curve, and hence infinitely many elliptic curves defined over '''Q''' with ''n''-torsion for these values of ''n''.  The converse statement, that only these values of ''n'' can occur, is [[Mazur's torsion theorem]].

=== ''X''<sub>0</sub>(''N'') of genus one ===

The modular curves <math>\textstyle X_0(N)</math> are of genus one if and only if <math>\textstyle N</math> equals one of the 12 values listed in the following table.<ref>{{cite book |editor-last1=Birch |editor-first1=Bryan |editor-last2=Kuyk |editor-first2=Willem |date=1975 |title=Modular functions of one variable IV |location=Berlin, Heidelberg |series=Lecture Notes in Mathematics |volume=476 |publisher=Springer-Verlag|page=79  |isbn=3-540-07392-2}}</ref> As [[elliptic curve]]s over <math>\mathbb{Q}</math>, they have minimal, integral Weierstrass models <math>y^2 + a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6</math>. This is, <math>\textstyle a_j\in\mathbb{Z}</math> and the absolute value of the discriminant <math>\Delta</math> is minimal among all integral Weierstrass models for the same curve. The following table contains the unique ''reduced'', minimal, integral Weierstrass models, which means <math>\textstyle a_1, a_3\in\{0,1\}</math> and <math>\textstyle a_2\in\{-1,0,1\}</math>.<ref>{{cite journal |last1=Ligozat |first1=Gerard |date=1975 |title=Courbes modulaires de genre 1 |url=http://www.numdam.org/article/MSMF_1975__43__5_0.pdf |journal=Bulletin de la Société Mathématique de France |volume=43 |issue= |pages=44–45 |access-date=2022-11-06}}</ref> The last column of this table refers to the home page of the respective elliptic modular curve <math>\textstyle X_0(N)</math> on ''[[The L-functions and modular forms database (LMFDB)]]''. 
{| class="wikitable"
|+ <math>X_0(N)</math> of genus 1
|-
! colspan="4"| <math>y^2 + a_1 x y + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6</math>
|-
|<math>N</math> || <math>[a_1,a_2,a_3,a_4,a_6]</math> || <math>\Delta</math> || LMFDB
|-
| 11 || [0, -1, 1, -10, -20] || <math>\textstyle -11^5</math> || [https://www.lmfdb.org/EllipticCurve/Q/11a1/ link]
|-
| 14 || [1, 0, 1, 4, -6] || <math>\textstyle -2^6\cdot 7^3</math> || [https://www.lmfdb.org/EllipticCurve/Q/14a1/ link]
|-
| 15 || [1, 1, 1, -10, -10] || <math>\textstyle 3^4\cdot 5^4</math> || [https://www.lmfdb.org/EllipticCurve/Q/15a1/ link]
|-
| 17 || [1, -1, 1, -1, -14] || <math>\textstyle -17^4</math> || [https://www.lmfdb.org/EllipticCurve/Q/17a1/ link]
|-
| 19 || [0, 1, 1, -9, -15] || <math>\textstyle -19^3</math> || [https://www.lmfdb.org/EllipticCurve/Q/19a1/ link]
|-
| 20 || [0, 1, 0, 4, 4] || <math>\textstyle -2^8\cdot 5^2</math> || [https://www.lmfdb.org/EllipticCurve/Q/20a1/ link]
|-
| 21 || [1, 0, 0, -4, -1] || <math>\textstyle 3^4\cdot 7^2</math> || [https://www.lmfdb.org/EllipticCurve/Q/21a1/ link]
|-
| 24 || [0, -1, 0, -4, 4] || <math>\textstyle 2^8\cdot 3^2</math> || [https://www.lmfdb.org/EllipticCurve/Q/24a1/ link]
|-
| 27 || [0, 0, 1, 0, -7] || <math>\textstyle -3^9</math> || [https://www.lmfdb.org/EllipticCurve/Q/27a1/ link]
|-
| 32 || [0, 0, 0, 4, 0] || <math>\textstyle -2^{12}</math> || [https://www.lmfdb.org/EllipticCurve/Q/32a1/ link]
|-
| 36 || [0, 0, 0, 0, 1] || <math>\textstyle -2^4\cdot 3^3</math> || [https://www.lmfdb.org/EllipticCurve/Q/36a1/ link]
|-
| 49 || [1, -1, 0, -2, -1] || <math>\textstyle -7^3</math> || [https://www.lmfdb.org/EllipticCurve/Q/49a1/ link]
|}