Values of the elliptic nome $q(m)$

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Numbers

$m$

$q(m)$

1/12:

0.005437996640958888763604333944263244362218248226319487081175973574773308701681748615685389822331061925

1/11:

0.005956604432645852138995816837682580301185317157025760150192424812803992896312343742653645085356027182

1/10:

0.006584651553858370274473059670654416375794733511909831916783096060362189022172615953412669252189503792

1/9:

0.007360907921497918897229857196946893712477130933441642711709290090580125906700332678040896466428442521

1/8:

0.008344937157754231533191962224202635534246809500651476466019143452386301304222209234269561297985501667

1/7:

0.009633225450975077847993324819370023076857051673013111944972884396718583454182116071023241990651833538

1/6:

0.01139312524885567543954049194589730663798459167951468602399481234413196179143066532951343676899018227

2/11:

0.01253928929458714978073488295670514604930058047446921110323727098883024449519300016606427573643945492

1/5:

0.01394285727531826872146408553480080836245826192308997930285905524339184933511297300620046993781034352

2/9:

0.01570198880343212250568593615240619007513802544765264721454873396072801391933439632143932479463821280

1/4:

0.01797238700896723999881969294898232019756170895758890494600032136730302838435227167942023020945237494

3/11:

0.01989285829860665545679689744248154209815438061302704964314143067600781825230242902250450027199812125

2/7:

0.02101713162097052939571160622760813044902535284967048306340636671036832317708665451954603936240976338

3/10:

0.02227743615715350822901627011198294308947627130982326852871835215961679594490279633083742431871085539

1/3:

0.02531991336628518686662697111068020250404898773186718007525917907544867263492388015091115260204815270

4/11:

0.02821908696450146422493866538836493795243562935917603293950363589070251186534489141807614601925486126

3/8:

0.02934151958781259016214996288792121823177508358295712970504165549322536571657314090816186631006210764

2/5:

0.03188334731336317755064298984250651892146924453021384219239621312299643280713473750908120731583520661

5/12:

0.03363648721310832052684319398526434972440719764739491595746513140105759882027080204818625497796227564

3/7:

0.03491915294137460904407690555379640754514021518987881502714883772240913427995154357906649757605874423

4/9:

0.03667083605803632060553408337478802607506779354128150600111641955554485129399019341610385052185618762

5/11:

0.03781131661777864479473905636873234899645528517112328604755483974999120422812535312346973984759913059

1/2:

0.04321391826377224977441773717172801127572810981063308298071968740105076575701796769813995996190108439

6/11:

0.04912021833762966793607640552195967390466322449109316744271223932521560173387114595323584451660952361

5/9:

0.05051091479758284580599184853973223331899866941955626837595878837966400933740288063957849921323842055

4/7:

0.05275980831811056980221382805442847804459408350092589578135344339182204704594288862690343167991387473

7/12:

0.05450037482243395278980631215325731010189492718419077354738014373599659555273570548100465962955388153

3/5:

0.05702025781460967637754953397628972554904223787533516384443350030918921743366045120912826528339978048

5/8:

0.06099819684361545026922536637079238106906389277494619619167591134144609441097568381008151022083581744

7/11:

0.06289218449274117268557442990648598823901786697698705798851424171376695845180801962872932788629079579

2/3:

0.06823782774533833308838151682774606046190537952604313036935405587308951238163800727142321538748023722

7/10:

0.07468994353717944761143751491387024830871777654149148411161300569422510657440679375273633875974612158

5/7:

0.07766954596395487611760544932061504690409640066510557959551735623252918765950377708880595601331430588

8/11:

0.08050541607105305401674777471176323741946364025862323768881283298572640251126261523074146151123109684

3/4:

0.08579573370219476651684045332013664996846941513985733149124984247598156209298196835733071519913580374

7/9:

0.09292651703816638553875269288842813011085313986450384435710454620808413367255150390980474890664923767

4/5:

0.09927369733882489703607378169883902447208532277216978370306400499817763640609872790914676155886758809

9/11:

0.1049883018051967424679660769695129381655871950888785574083390978080057205771296832184799619878408351

5/6:

0.1101817726981566362935596029647893662548184786257184893317810180874035348769407271966474517269821499

6/7:

0.1193246369154182051032915727712712579953551069733134256808678882611735921357077304471090181511517800

7/8:

0.1271816262512603744280631851049683473528150048020029884880147871679251283154495636891886903524432362

8/9:

0.1340611316115422024331415926671720727725739710000142776548368536040383191939663463767340575208483947

9/10:

0.1401731269542615524091054546101984273550836589486169206788087430985991845766228482507284055186913259

10/11:

0.1456669375144607936816650401457592457171780304771229777386991754434199642366459270925802548330891607

11/12:

0.1506524897868949459144556396602432959717248465620969128773693408273844022693719717428309134141609551

Definition

For $0<m<1$, this table stores the real elliptic nome $q(m)=\exp(-\pi K(1-m)/K(m))$, where $K$ is Legendre's $K$ in the parameter convention $m=k^2$ [2] [3].

Parameters

$m$

— elliptic parameter ($0<m<1$)

Formulas

(1)

If $\tau=iK(1-m)/K(m)$, then $q(m)=e^{i\pi\tau}$.

(2)

With the same nome, $m=\theta_2(0,q(m))^4/\theta_3(0,q(m))^4$ [2].

(3)

$q(1/2)=e^{-\pi}$.

Comments

(4)

The stored parameter is $m=k^2$, matching the complete first-kind elliptic integral table and Sage's elliptic integral convention. A source that writes the nome as a function of the modulus $k$ is using this table's $q(k^2)$.

(5)

The nome convention here is the theta-function convention $q=e^{i\pi\tau}$ [1]. Some modular-form sources write $q$ for $e^{2\pi i\tau}$, which is the square of this nome [4].

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.complex_arb import ComplexBallField
from sage.rings.rational_field import QQ

CBF = ComplexBallField(numberdb.bits(100, losing=80))
m = CBF(QQ(1) / QQ(2))
q = (-CBF.pi() * (CBF(1) - m).elliptic_k() / m.elliptic_k()).exp()
q.real()

Links

[1]

DLMF 20.1: Special notation for theta functions

[2]

DLMF 20.9: Relations of theta functions to elliptic functions

[3]

DLMF 19.2: Definitions of elliptic integrals

[4]

MathWorld: Nome

Similar tables

Complete elliptic integral of the first kind $K(m)$ — $K(m)$ and $K(1-m)$ are the two complete first-kind integrals in the defining quotient

Singular values $k_r$ of the elliptic modulus — for the singular parameter $m_r$ stored there, $K(1-m_r)/K(m_r)=\sqrt r$ and $q(m_r)=e^{-\pi\sqrt r}$

Complete elliptic integral of the first kind $K(m_r)$ at the singular values — holds the $K(m_r)$ factor at the singular parameters for which the nome is $e^{-\pi\sqrt r}$

Values of the Rogers-Ramanujan continued fraction $R(e^{-\pi\sqrt r})$ — evaluates another function at the singular nomes $e^{-\pi\sqrt r}$

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds $q(m)$ at every rational $m=a/b$ in lowest terms with $b\leq12$ and $0<m<1$, which are the arguments a reader is likely to have written down rather than the ones base ten makes short; $m=1/2$ is the distinguished one, where $q=e^{-\pi}$)

How they were obtained:

Each value was computed as a Sage complex ball with arb at numberdb.bits(digits, losing=80) bits, with the imaginary part checked to contain zero, and returned as its real part. The run checked (3) and (2) across the stored range.