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ID: INPUT{id.yaml}

Title: >
  Modular polynomials for Weber's $f$-function

Definition: >
  This table contains the modular polynomials
  $\phi_\ell(x,y) \in \mathbb{Z}[x,y]$ for some small primes $\ell \geq 5$, 
  which vanish on the points $(f(\ell\tau),f(\tau))$, where $f$ denotes
  the Weber $f$-function.
  
Parameters:
  ell:
    display: $\ell$
    type: Z
    constraints: $\ell \geq 5$ (prime)
   
Comments:
  comment-relation-to-j: >
    Weber's $f$-function is related to the $j$-invariant via
    $j = (f^{24}-16)^3/f^{24}$.
  comment-more-data: >
    More complete data is available at CITE{Suth}.
  
Formulas:

Programs:

References:
  BrLaSu:
    bib: >
      Reinier Broker, Kristin Lauter, Andrew V. Sutherland,
      "Modular polynomials via isogeny volcanoes",
      Mathematics of Computation 81 (2012), 1201-1231.
    arXiv: 1001.0402
  BrOnSu:
    bib: >
      Jan Hendrik Bruinier, Ken Ono, Andrew V. Sutherland,
      "Class polynomials for nonholomorphic modular functions",
      Journal of Number Theory 161 (2016) 204-229.
    arXiv: 1301.5672

Links:
  Suth:
    title: "A. V. Sutherland: Modular polynomials for Weber's f-function" 
    url: https://math.mit.edu/~drew/WeberModPolys.html
  Wiki:
    title: "Wikipedia: Weber modular function"
    url: https://en.wikipedia.org/wiki/Weber_modular_function

Similar tables:
  
Keywords:
  
Tags:
- number theory
- elliptic curves
- modular function

Data properties:
  type: Z[]
  complete: no
  sources:
  - CITE{Suth}
  - CITE{BrLaSu}
  - CITE{BrOnSu}
 
Display properties:
  number-header: $\phi_\ell(x,y)$

Numbers: INPUT{polynomials.yaml}

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Modular polynomials for Weber's $f$-function

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Numbers

$\ell$

$\phi_\ell(x,y)$

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

Definition

This table contains the modular polynomials $\phi_\ell(x,y) \in \mathbb{Z}[x,y]$ for some small primes $\ell \geq 5$, which vanish on the points $(f(\ell\tau),f(\tau))$, where $f$ denotes the Weber $f$-function.

Parameters

$\ell$

— integer ($\ell \geq 5$ (prime))

Comments

(1)

Weber's $f$-function is related to the $j$-invariant via $j = (f^{24}-16)^3/f^{24}$.

(2)

More complete data is available at [3].

References

[1]

Reinier Broker, Kristin Lauter, Andrew V. Sutherland, "Modular polynomials via isogeny volcanoes", Mathematics of Computation 81 (2012), 1201-1231. (arXiv)

[2]

Jan Hendrik Bruinier, Ken Ono, Andrew V. Sutherland, "Class polynomials for nonholomorphic modular functions", Journal of Number Theory 161 (2016) 204-229. (arXiv)

Links

[3]

A. V. Sutherland: Modular polynomials for Weber's f-function

[4]

Wikipedia: Weber modular function

Data properties

Entries are of type: integral polynomial

Table is complete: no

Sources of data: [3], [1], [2]