Editing Modular curve (section)

=== ''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]
|}