Benutzer:Chrgue/X0(N)

The modular curves ${\displaystyle \textstyle X_{0}(N)}$ are of genus 1 if and only if ${\displaystyle \textstyle N}$ equals one of the 12 values listed in the following table.^[1] As elliptic curves over ${\displaystyle \mathbb {Q} }$, they have minimal, integral Weierstrass models ${\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}}$. This is, ${\displaystyle \textstyle a_{j}\in \mathbb {Z} }$ and the absolute value of the discriminant ${\displaystyle \Delta }$ is minimal among all integral Weierstrass models for the same curve. The following table contains the unique reduced, minimal, integral Weierstrass models, which means ${\displaystyle \textstyle a_{1},a_{3}\in \{0,1\}}$ and ${\displaystyle \textstyle a_{2}\in \{-1,0,1\}}$, for the modular curves ${\displaystyle \textstyle X_{0}(N)}$ of genus 1.^[2] The last column of this table refers to the home page of the respective elliptic modular curve ${\displaystyle \textstyle X_{0}(N)}$ on The L-functions and modular forms database (LMFDB).

${\displaystyle X_{0}(N)}$ of genus 1

${\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}}$
${\displaystyle N}$	${\displaystyle [a_{1},a_{2},a_{3},a_{4},a_{6}]}$	${\displaystyle \Delta }$	LMFDB
11	[0, -1, 1, -10, -20]	${\displaystyle \textstyle -11^{5}}$	link
14	[1, 0, 1, 4, -6]	${\displaystyle \textstyle -2^{6}\cdot 7^{3}}$	link
15	[1, 1, 1, -10, -10]	${\displaystyle \textstyle 3^{4}\cdot 5^{4}}$	link
17	[1, -1, 1, -1, -14]	${\displaystyle \textstyle -17^{4}}$	link
19	[0, 1, 1, -9, -15]	${\displaystyle \textstyle -19^{3}}$	link
20	[0, 1, 0, 4, 4]	${\displaystyle \textstyle -2^{8}\cdot 5^{2}}$	link
21	[1, 0, 0, -4, -1]	${\displaystyle \textstyle 3^{4}\cdot 7^{2}}$	link
24	[0, -1, 0, -4, 4]	${\displaystyle \textstyle 2^{8}\cdot 3^{2}}$	link
27	[0, 0, 1, 0, -7]	${\displaystyle \textstyle -3^{9}}$	link
32	[0, 0, 0, 4, 0]	${\displaystyle \textstyle -2^{12}}$	link
36	[0, 0, 0, 0, 1]	${\displaystyle \textstyle -2^{4}\cdot 3^{3}}$	link
49	[1, -1, 0, -2, -1]	${\displaystyle \textstyle -7^{3}}$	link

1. ↑ Bryan Birch, Willem Kuyk (Ed.): Modular functions of one variable IV. 1975, Lecture Notes in Mathematics, Band 476, Springer-Verlag, ISBN 3-540-07392-2, S. 79.
2. ↑ Gerard Ligozat: Courbes modulaires de genre 1. 1975, Bulletin de la Société Mathématique de France, Mémoire 43, Abschnitt (4.2.6), S. 45.