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ID: INPUT{id.yaml}

Title: >
  Regulators of elliptic curves over $\mathbb{Q}$ of rank $1$

Definition: >
  The regulator of an elliptic curve $E$ over $\mathbb{Q}$ is the 
  determinant of the height pairing matrix for a
  Mordell--Weil basis of $E(\mathbb{Q})$.
  For elliptic curves of rank $1$, this is the same as the
  Néron-Tate height of a Mordell-Weil generator of the free part 
  of $E(\mathbb{Q})$.
  This table lists regulators for elliptic curves over $\mathbb{Q}$ of
  rank $1$ 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-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:
  formula-height-pairing-matrix: >
    For an elliptic curve $E$ over a number field $K$, 
    $\text{Reg}(E/K) = \det(\langle P_i, P_j\rangle)_{1\leq i, j \leq r}$, 
    where $P_1, \ldots, P_r$ is a Mordell-Weil basis for $E(K)$ and
    $\langle ., .\rangle$ denotes the Néron-Tate height pairing.

Programs:
  program-sage: 
    language: Sage
    code: |
      numbers = [E.regulator() for E in cremona_curves([1..100])
                               if E.rank() == 1]

References:
      
Links:
  LMFDB:
    title: "LMFDB: Elliptic curves over Q"
    url: https://www.lmfdb.org/EllipticCurve/Q/
  Wiki:
    title: "Wikipedia: Elliptic regulator"
    url: https://en.wikipedia.org/wiki/N%C3%A9ron%E2%80%93Tate_height#The_elliptic_and_abelian_regulators
    
Similar tables:
  
Keywords:
  
Tags:
- number theory
- elliptic curves

Data properties:
  type: R
  complete: unknown
  sources:
  - CITE{LMFDB}

Display properties:
  number-header: $\text{Reg}(E/\mathbb{Q})$
  group parameters:
  - [N]
  - [c4, c6]

Numbers: INPUT{numbers.yaml}

— preview —

Regulators of elliptic curves over $\mathbb{Q}$ of rank $1$

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Numbers

$N$

$c_4$, $c_6$

$\text{Reg}(E/\mathbb{Q})$

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

Definition

The regulator of an elliptic curve $E$ over $\mathbb{Q}$ is the determinant of the height pairing matrix for a Mordell--Weil basis of $E(\mathbb{Q})$. For elliptic curves of rank $1$, this is the same as the Néron-Tate height of a Mordell-Weil generator of the free part of $E(\mathbb{Q})$. This table lists regulators for elliptic curves over $\mathbb{Q}$ of rank $1$ 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 $E$

Formulas

(1)

For an elliptic curve $E$ over a number field $K$, $\text{Reg}(E/K) = \det(\langle P_i, P_j\rangle)_{1\leq i, j \leq r}$, where $P_1, \ldots, P_r$ is a Mordell-Weil basis for $E(K)$ and $\langle ., .\rangle$ denotes the Néron-Tate height pairing.

Comments

(2)

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

(3)

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.regulator() for E in cremona_curves([1..100])
                         if E.rank() == 1]

Links

[1]

LMFDB: Elliptic curves over Q

[2]

Wikipedia: Elliptic regulator

Data properties

Entries are of type: real number

Table is complete: unknown

Sources of data: [1]