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ID: INPUT{id.yaml}

Title: >
  Special $L$-value of elliptic curves over $\mathbb{Q}$ of rank $3$

Definition: >
  The special $L$-value of an elliptic curve over $\mathbb{Q}$ 
  of (algebraic) rank $r$ is 
  the $r$'th derivative of 
  its $L$-function $L(E,s)$ at $s=1$.
  Here the arithmetic normalization of $L(E,s)$ is used CITE{LMFDB-special-L-value}.
  This table lists $L^{(3)}(E,1)$ for elliptic curves over $\mathbb{Q}$ of
  rank $3$ for the first few conductors.
  
Parameters:
  N:
    title: conductor of $E$
    type: Z
  c4:
    display: $c_4$
    #title: HREF{https://www.lmfdb.org/knowledge/show/ec.q.lmfdb_label}[LMFDB label]
    title: invariant $c_4$ of a minimal model of $E$
    type: Z

c6:
    display: $c_6$
    title: invariant $c_6$ of a minimal model of $E$
    type: Z
   
Comments:
  comment-optimal-curves-only: >
    As $L(E,s)$ only depends on the $\mathbb{Q}$-isogeny class of $E$,
    the table lists at most one representative of each isogeny class.
  comment-BSD: >
    The Birch and Swinnerton-Dyer conjecture CITE{WikiBSD} states that
    $\frac{L^{(r)}(E,1)}{r!} = \frac{|\text{Sha}(E)|\Omega_E R_E \prod_{p|N} c_p}{|E_{\text{Tor}}|^2}$.
  comment-redundancy-of-N: >
    The conductor $N$ can be deduced from $c_4$ and $c_6$.
    We display it for convenience.
  comment-E-from-c4c6: >
    A model of $E/\mathbb{Q}$ is given by $E: y^2 = x^3 - 27 c_4 x - 54 c_6$.
    
Formulas:

Programs:
  program-sage: 
    language: Sage
    code: |
      numbers = [E.lseries().taylor_series(series_prec=4)[3] * factorial(3) 
                               for E in cremona_curves([1..100000])
                               if E.rank() == 3]

References:
      
Links:
  LMFDB:
    title: "LMFDB: Elliptic curves over Q"
    url: https://www.lmfdb.org/EllipticCurve/Q/
  LMFDB-special-L-value:
    title: "LMFDB: Special value of an elliptic curve L-function"
    url: https://www.lmfdb.org/knowledge/show/ec.q.special_value?special_value
  WikiBSD:
    title: "Wikipedia: Birch and Swinnerton-Dyer conjecture"
    url: https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture
    
Similar tables:
  
Keywords:
  
Tags:
- number theory
- elliptic curves
- special values
- L-function

Data properties:
  type: R
  complete: no
  sources:
  - CITE{LMFDB}

Display properties:
  number-header: $L^{(3)}(s,1)$
  group parameters:
  - [N]
  - [c4, c6]

Numbers: INPUT{numbers.yaml}

— preview —

Special $L$-value of elliptic curves over $\mathbb{Q}$ of rank $3$

edit on github

Numbers

$N$

$c_4$, $c_6$

$L^{(3)}(s,1)$

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

Definition

The special $L$-value of an elliptic curve over $\mathbb{Q}$ of (algebraic) rank $r$ is the $r$'th derivative of its $L$-function $L(E,s)$ at $s=1$. Here the arithmetic normalization of $L(E,s)$ is used [2]. This table lists $L^{(3)}(E,1)$ for elliptic curves over $\mathbb{Q}$ of rank $3$ for the first few conductors.

Parameters

$N$

— conductor of $E$

$c_4$

— invariant $c_4$ of a minimal model of $E$

$c_6$

— invariant $c_6$ of a minimal model of $E$

Comments

(1)

As $L(E,s)$ only depends on the $\mathbb{Q}$-isogeny class of $E$, the table lists at most one representative of each isogeny class.

(2)

The Birch and Swinnerton-Dyer conjecture [3] states that $\frac{L^{(r)}(E,1)}{r!} = \frac{|\text{Sha}(E)|\Omega_E R_E \prod_{p|N} c_p}{|E_{\text{Tor}}|^2}$.

(3)

The conductor $N$ can be deduced from $c_4$ and $c_6$. We display it for convenience.

(4)

A model of $E/\mathbb{Q}$ is given by $E: y^2 = x^3 - 27 c_4 x - 54 c_6$.

Programs

(P1)

Sage

numbers = [E.lseries().taylor_series(series_prec=4)[3] * factorial(3) 
                         for E in cremona_curves([1..100000])
                         if E.rank() == 3]

Links

[1]

LMFDB: Elliptic curves over Q

[2]

LMFDB: Special value of an elliptic curve L-function

[3]

Wikipedia: Birch and Swinnerton-Dyer conjecture

Data properties

Entries are of type: real number

Table is complete: no

Sources of data: [1]