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ID: INPUT{id.yaml}

Title: >
  Rational singular moduli

Definition: >
  Let $H$ be the upper half plane and $j$ the $j$-invariant.
  A singular modulus is a number of the form $j(\tau)$ 
  for an imaginary quadratic $\tau \in H$.
  This list contains all rational singula moduli.
  
Parameters:
  Delta:
    display: $\Delta$
    title: Discriminant
    type: Z
   
Comments:
  comment-discriminant: >
    $\Delta$ is the associated discriminant of the 
    complex multiplication order 
    $\mathcal{O} = \text{End}\langle\tau,1\rangle$.
        
Formulas:

Programs:
  program-sage: 
    language: Sage
    code: |
      numbers = {d*f^2: j for d, f, j in cm_j_invariants_and_orders(QQ)}

References:
      
Links:
  Wiki:
    title: "Wikipedia: Singular moduli"
    url: https://en.wikipedia.org/wiki/Complex_multiplication#Singular_moduli
        
Similar tables:
  
Keywords:
  
Tags:
- number theory
- elliptic curves

Data properties:
  type: Z
  complete: yes

Display properties:
  number-header: $j$

Numbers: INPUT{numbers.yaml}

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Rational singular moduli

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Numbers

$\Delta$

$j$

INPUT{numbers.yaml} (not shown in preview)

Definition

Let $H$ be the upper half plane and $j$ the $j$-invariant. A singular modulus is a number of the form $j(\tau)$ for an imaginary quadratic $\tau \in H$. This list contains all rational singula moduli.

Parameters

$\Delta$

— Discriminant

Comments

(1)

$\Delta$ is the associated discriminant of the complex multiplication order $\mathcal{O} = \text{End}\langle\tau,1\rangle$.

Programs

(P1)

Sage

numbers = {d*f^2: j for d, f, j in cm_j_invariants_and_orders(QQ)}

Links

[1]

Wikipedia: Singular moduli

Data properties

Entries are of type: integer

Table is complete: yes