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Regulators of elliptic curves over real quadratic fields
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Numbers
$D$
label 
$R_{E/K}$
12
47.1-b1:
0.098568841514602893460165344842850895
comment: LMFDB curve 2.2.12.1-47.1-b1 has conductor ideal $(-4w-1)$, rank $1$, and equation ${y}^2+{x}{y}+{y}={x}^{3}+w{x}^{2}+\left(w-1\right){x}$.
12
47.1-b2:
0.049284420757301446730082672421425447
comment: LMFDB curve 2.2.12.1-47.1-b2 has conductor ideal $(-4w-1)$, rank $1$, and equation ${y}^2+{x}{y}+{y}={x}^{3}+w{x}^{2}+\left(6w-11\right){x}-14w+24$.
12
47.2-b1:
0.098568841514602893460165344842850895
comment: LMFDB curve 2.2.12.1-47.2-b1 has conductor ideal $(4w-1)$, rank $1$, and equation ${y}^2+{x}{y}+{y}={x}^{3}-w{x}^{2}+\left(-w-1\right){x}$.
12
47.2-b2:
0.049284420757301446730082672421425447
comment: LMFDB curve 2.2.12.1-47.2-b2 has conductor ideal $(4w-1)$, rank $1$, and equation ${y}^2+{x}{y}+{y}={x}^{3}-w{x}^{2}+\left(-6w-11\right){x}+14w+24$.
13
51.2-a1:
0.034450165062797493363476807161562966
comment: LMFDB curve 2.2.13.1-51.2-a1 has conductor ideal $(-2w+9)$, rank $1$, and equation ${y}^2+w{y}={x}^{3}-{x}^{2}+\left(-2w+4\right){x}+w-3$.
13
51.2-a2:
0.011483388354265831121158935720520988
comment: LMFDB curve 2.2.13.1-51.2-a2 has conductor ideal $(-2w+9)$, rank $1$, and equation ${y}^2+w{y}={x}^{3}-{x}^{2}+\left(w-1\right){x}-w+1$.
13
51.3-a1:
0.011483388354265831121158935720520988
comment: LMFDB curve 2.2.13.1-51.3-a1 has conductor ideal $(2w+7)$, rank $1$, and equation ${y}^2+\left(w+1\right){y}={x}^{3}-{x}^{2}-w{x}$.
13
51.3-a2:
0.034450165062797493363476807161562966
comment: LMFDB curve 2.2.13.1-51.3-a2 has conductor ideal $(2w+7)$, rank $1$, and equation ${y}^2+\left(w+1\right){y}={x}^{3}-{x}^{2}+\left(2w+2\right){x}-2w-2$.
13
52.1-b1:
1.3834480278188076609662078599889399
comment: LMFDB curve 2.2.13.1-52.1-b1 has conductor ideal $(-4w+2)$, rank $1$, and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-213{x}-1257$.
13
52.1-b2:
0.19763543254554395156660112285556285
comment: LMFDB curve 2.2.13.1-52.1-b2 has conductor ideal $(-4w+2)$, rank $1$, and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-3{x}+3$.
13
53.1-a1:
0.047438850434816421561347347750918711
comment: LMFDB curve 2.2.13.1-53.1-a1 has conductor ideal $(w+7)$, rank $1$, and equation ${y}^2+w{x}{y}+{y}={x}^{3}-w{x}^{2}+\left(-w+1\right){x}$.
13
53.2-a1:
0.047438850434816421561347347750918711
comment: LMFDB curve 2.2.13.1-53.2-a1 has conductor ideal $(w-8)$, rank $1$, and equation ${y}^2+\left(w+1\right){x}{y}+{y}={x}^{3}-{x}^{2}$.
17
17.1-a1:
6.0705123554606643216641072065434946
comment: LMFDB curve 2.2.17.1-17.1-a1 has conductor ideal $(-2w+1)$, rank $1$, and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(330w-936\right){x}+4996w-13110$.
17
17.1-a2:
0.75881404443258304020801340081793683
comment: LMFDB curve 2.2.17.1-17.1-a2 has conductor ideal $(-2w+1)$, rank $1$, and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-{x}-14$.
17
17.1-a3:
0.37940702221629152010400670040896841
comment: LMFDB curve 2.2.17.1-17.1-a3 has conductor ideal $(-2w+1)$, rank $1$, and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(6w+9\right){x}+4w+6$.
17
17.1-a4:
0.75881404443258304020801340081793683
comment: LMFDB curve 2.2.17.1-17.1-a4 has conductor ideal $(-2w+1)$, rank $1$, and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-{x}$.
17
17.1-a5:
1.5176280888651660804160268016358736
comment: LMFDB curve 2.2.17.1-17.1-a5 has conductor ideal $(-2w+1)$, rank $1$, and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-6{x}-4$.
17
17.1-a6:
0.37940702221629152010400670040896841
comment: LMFDB curve 2.2.17.1-17.1-a6 has conductor ideal $(-2w+1)$, rank $1$, and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-6w+15\right){x}-4w+10$.
17
17.1-a7:
3.0352561777303321608320536032717473
comment: LMFDB curve 2.2.17.1-17.1-a7 has conductor ideal $(-2w+1)$, rank $1$, and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}-91{x}-310$.
17
17.1-a8:
6.0705123554606643216641072065434946
comment: LMFDB curve 2.2.17.1-17.1-a8 has conductor ideal $(-2w+1)$, rank $1$, and equation ${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+\left(-330w-606\right){x}-4996w-8114$.
17
38.2-a1:
0.085387436507748877571154453414499536
comment: LMFDB curve 2.2.17.1-38.2-a1 has conductor ideal $(-3w+2)$, rank $1$, and equation ${y}^2+{x}{y}={x}^{3}+\left(-w+1\right){x}^{2}+\left(3w-7\right){x}-2w+5$.
17
38.2-a2:
0.17077487301549775514230890682899907
comment: LMFDB curve 2.2.17.1-38.2-a2 has conductor ideal $(-3w+2)$, rank $1$, and equation ${y}^2+{x}{y}+w{y}={x}^{3}+{x}^{2}+\left(-4w-5\right){x}+w+1$.
17
38.2-a3:
0.34154974603099551028461781365799814
comment: LMFDB curve 2.2.17.1-38.2-a3 has conductor ideal $(-3w+2)$, rank $1$, and equation ${y}^2+{x}{y}+\left(w+1\right){y}={x}^{3}+\left(w-5\right){x}+w-6$.
17
38.2-a4:
0.085387436507748877571154453414499536
comment: LMFDB curve 2.2.17.1-38.2-a4 has conductor ideal $(-3w+2)$, rank $1$, and equation ${y}^2+{x}{y}+w{y}={x}^{3}+{x}^{2}+\left(-49w-75\right){x}+243w+379$.
17
38.3-a1:
0.085387436507748877571154453414499536
comment: LMFDB curve 2.2.17.1-38.3-a1 has conductor ideal $(3w-1)$, rank $1$, and equation ${y}^2+{x}{y}+\left(w+1\right){y}={x}^{3}+{x}^{2}+\left(48w-124\right){x}-244w+622$.
17
38.3-a2:
0.17077487301549775514230890682899907
comment: LMFDB curve 2.2.17.1-38.3-a2 has conductor ideal $(3w-1)$, rank $1$, and equation ${y}^2+{x}{y}+\left(w+1\right){y}={x}^{3}+{x}^{2}+\left(3w-9\right){x}-2w+2$.
17
38.3-a3:
0.085387436507748877571154453414499536
comment: LMFDB curve 2.2.17.1-38.3-a3 has conductor ideal $(3w-1)$, rank $1$, and equation ${y}^2+{x}{y}={x}^{3}+w{x}^{2}+\left(-3w-4\right){x}+2w+3$.
17
38.3-a4:
0.34154974603099551028461781365799814
comment: LMFDB curve 2.2.17.1-38.3-a4 has conductor ideal $(3w-1)$, rank $1$, and equation ${y}^2+{x}{y}+w{y}={x}^{3}+\left(-2w-3\right){x}-2w-4$.
17
47.1-a1:
0.047398169405353604132198011561955181
comment: LMFDB curve 2.2.17.1-47.1-a1 has conductor ideal $(-2w+9)$, rank $1$, and equation ${y}^2+{y}={x}^{3}+\left(w-1\right){x}^{2}+{x}$.
17
47.2-a1:
0.047398169405353604132198011561955181
comment: LMFDB curve 2.2.17.1-47.2-a1 has conductor ideal $(2w+7)$, rank $1$, and equation ${y}^2+{y}={x}^{3}-w{x}^{2}+{x}$.
17
52.3-b1:
0.011548536804434145224328626608586774
comment: LMFDB curve 2.2.17.1-52.3-b1 has conductor ideal $(w+7)$, rank $1$, and equation ${y}^2+\left(w+1\right){y}={x}^{3}+\left(-16w-25\right){x}+50w+78$.
17
52.6-b1:
0.011548536804434145224328626608586774
comment: LMFDB curve 2.2.17.1-52.6-b1 has conductor ideal $(w-8)$, rank $1$, and equation ${y}^2+w{y}={x}^{3}+\left(16w-41\right){x}-51w+129$.
Definition
Let $K=\mathbb{Q}(\sqrt{D})$ and let $E/K$ be an LMFDB-labelled elliptic curve of positive Mordell-Weil rank $r$. This table gives the regulator $R_{E/K}$, the determinant of the absolute Néron-Tate height pairing on $E(K)$ modulo torsion.
Parameters
$D$
—   field discriminant (positive fundamental discriminant)
label
—   LMFDB curve label, without the field prefix (the LMFDB curve label with the field label $2.2.D.1$ removed: 31.1-a1 for the curve 2.2.5.1-31.1-a1)
Formulas
(1)
The Birch and Swinnerton-Dyer formula [3], with the $L$-function normalised as in the Definition, is $L^*(E/K,1)=2^r\Omega_{E/K}R_{E/K}|\operatorname{Sha}(E/K)| \prod_{\mathfrak p}c_{\mathfrak p}/(|E(K)_{\mathrm{tors}}|^2\sqrt{D})$. Here $L^*(E/K,1)=L^{(r)}(E/K,1)/r!$ and the order of vanishing is $r$, as the conjecture predicts; $\Omega_{E/K}$ is the product, over the two real embeddings of $K$, of the real period of $E$ times the number of connected components of $E(\mathbb{R})$ at that embedding; $R_{E/K}$ is the regulator for the absolute Néron-Tate height; $\operatorname{Sha}(E/K)$ is the Tate-Shafarevich group; and $c_{\mathfrak p}$ is the Tamagawa number at $\mathfrak p$.
Comments
(2)
The label of a row is the LMFDB [2] curve label with the field label removed: the row with $D=12$ and label 47.1-b1 is the curve 2.2.12.1-47.1-b1, where 47.1 labels the conductor ideal, of norm $47$, b is the isogeny class and 1 the curve in it. There is one row per curve rather than one row per isogeny class, since the regulator is attached to the Mordell-Weil lattice of the curve. A curve and its Galois conjugate have the same regulator, so over $D=13$ the rows whose conductor ideals are labelled 51.2 and 51.3 repeat one another with the curve numbers exchanged. Curves in one isogeny class have regulators in a rational ratio; in the class 17.1-a over $D=17$ the ratios are powers of $2$.
(3)
The generator $w$ of $K$ follows the LMFDB [2] convention used in ecnf-data [1]: it satisfies $w^2-w-1=0$ for $D=5$, $w^2-2=0$ for $D=8$, $w^2-3=0$ for $D=12$, $w^2-w-3=0$ for $D=13$, and $w^2-w-4=0$ for $D=17$.
(4)
Rank $0$ curves are not listed; their regulator is the determinant of the empty height-pairing matrix, so it is $1$.
(5)
The height pairing is absolute. In PARI, ellheightmatrix over a quadratic field gives twice this height; the ecnf-data reg field and this table use the absolute convention that appears in (1).
(6)
If $P_1,\dots,P_r$ is a basis of $E(K)$ modulo torsion, then $R_{E/K}=\det(\langle P_i,P_j\rangle)_{1\leq i,j\leq r}$; changing the basis does not change this determinant.
Links
Similar tables
Regulators of elliptic curves over $\mathbb{Q}$ of rank $1$ —   the same height-pairing regulator for elliptic curves over $\mathbb{Q}$ of rank $1$
Regulators of elliptic curves over $\mathbb{Q}$ of rank $2$ —   the same height-pairing regulator for elliptic curves over $\mathbb{Q}$ of rank $2$
Regulators of elliptic curves over $\mathbb{Q}$ of rank $3$ —   the same height-pairing regulator for elliptic curves over $\mathbb{Q}$ of rank $3$
Special $L$-values of elliptic curves over real quadratic fields —   the $L^*(E/K,1)$ side of the Birch and Swinnerton-Dyer formula for the same ecnf-data curves
Regulators of real quadratic fields —   a different regulator, of the unit group of the base field
Data properties
Entries are of type: real number
How well the digits are known: heuristic
Sources of data: [1]
Table is complete: no (it holds every positive-rank curve in ecnf-data over the five real quadratic fields of smallest discriminant, $D\in\{5,8,12,13,17\}$, with conductor norm at most $60$)
How they were obtained:

Entries are transcribed from the ecnf-data reg field at commit 10b28418e80392032b106ea00e6c5aa109d28e7b and kept to $35$ significant digits, because equality checks among Galois-conjugate curves and same-regulator curves in this table show that the final source digits are not stable.

more

In this range every stored curve has rank $1$, and each regulator is compared with the source height of the recorded generator. The generator also checks that the Birch and Swinnerton-Dyer quotient in (1) gives $|\operatorname{Sha}(E/K)|$, the analytic order of the Tate-Shafarevich group recorded by ecnf-data for each source row.



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