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Values of the Jacobi elliptic function $\operatorname{sn}(u\mid m)$
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Numbers
$m$
$u$ 
$\operatorname{sn}(u\mid m)$
1/8
1/8:
0.1246344905113571993734661956199447800993635093411658038154864964818523020706674964992250760990546442
1/8
1/6:
0.1658015608583328119088705037861222784331022461532371451548024509845641312698997621984106645320688815
1/8
1/4:
0.2470925797001463757910150288124658583565318917196657554979772991844446770060430965926400424197300121
1/8
1/3:
0.3264819521152859866931151510331945867535364888131815064081344699426788966140486478432927450559733689
1/8
1/2:
0.4772529886522666689938022959443638584971598527544119123408989838963369876717887341792210260312371080
1/8
2/3:
0.6139302174246066535136806526612900171607596578003710546572151141732215620390268287953795056963316218
1/8
3/4:
0.6758842384972414497201826468272640693233176779321330767093762612800478428848500262220024906971108757
1/8
1:
0.8321569101972313828766425015872174785209704502000396952740098928583027153607154360371446736549604027
1/8
5/4:
0.9391921636696641219490010345133571663122728988917055302095052692509069244849990173946665003616154777
1/8
4/3:
0.9632214905286049551172909673760386039695273463878366231173052749204190243650980361248925942391385695
1/8
3/2:
0.9933123294311842202603437016085694002557138299646130336612157211264711924053746870420700745015792500
1/8
5/3:
0.9991911102470012861370398799920999968327220398139755407902804326345394168360376818168019216401837145
1/8
7/4:
0.9930209555477373170059371843613556681042792417335222772791056576253531334681232581347320066423886169
1/8
2:
0.9383254059889736234433810597228028428235846763274835912032700812911056635339862910112321385981538883
1/8
9/4:
0.8307407986808553266775767430127296022164290403456780131932074294826427595203070651842964546062094820
1/8
7/3:
0.7836839828190686923527822233297898895009629152802121076537684818996974437615056270416550366338349030
1/8
5/2:
0.6739734323232577071249029475932771866108364974006914409536183201439020552171883069151748036601929052
1/8
8/3:
0.5453948163236302115876707987064670445743009605962474720600711550002984922664372912117848202242955390
1/8
11/4:
0.4749414484301690461077916203786230741824914802441284143033813714278483513872505544103826648722648674
1/8
3:
0.2445175589837799976376042063893443627361276558760622233509610053224554990890282985296474179428096579
1/6
1/8:
0.1246210778980693727771489703704133756188770362132071534831775588305417124975820169687532972588208634
1/6
1/6:
0.1657700437005473291329756716408467840657877574566746820495446528744913059770763344601508699585393882
1/6
1/4:
0.2469888320688772868636928813154883678342899175025975378340177953264638519917285058714035149590449926
1/6
1/3:
0.3262445289366347247910792465442152707242880594494024902159008571443304610337753223278555390522910778
1/6
1/2:
0.4765293584225316010744359647722239666788611478394767785462184381500877163106474131741704798069233858
1/6
2/3:
0.6124495838521394301762284426513517326528262351294618092518542847930627280657959150084991642785248283
1/6
3/4:
0.6739619237049240829725376326744576184720110233406397722616475129333706580879373562864342919405120528
1/6
1:
0.8290057889975734997643281725407268844208015488691564250427918138029105872287711476055712004781109071
1/6
5/4:
0.9357409558167278078450764465384603890534468652736318365753755192671030798610106721786558958341584904
1/6
4/3:
0.9600519103279681665846947197613032341010803740193962374778098048553652575782936498248216426176388013
1/6
3/2:
0.9914585585754713999251531231961130024181600522405869401450071890257191042163437007071985437600657238
1/6
5/3:
0.9997704937857180097185348397047178809862535579829855604376702127108093027823844282934471882527528776
1/6
7/4:
0.9952479061263738786074408597701786577826932367139699945636666592995504209981019790478292868022896477
1/6
2:
0.9470604056668158707614896444033786325619207418742292615097051043611816584395658251944753481014397824
1/6
9/4:
0.8476411104493078369944734064160170459790057695277642336355393033975746865957511580343764243862996895
1/6
7/3:
0.8034529194177833976451809015647361384800904727453785937995430878432899061719616228309184729818001000
1/6
5/2:
0.6993538832580258171168191293308983088225304274734059338020591667079357154037675383297792873154862103
1/6
8/3:
0.5758302612819459467075014427747008513918166986655225856020329301484412678918036481799073800972147545
1/6
11/4:
0.5075636427077397354451027142731893737481716316174100172285029780349414370018288437136861937828060349
1/6
3:
0.2818797811713201541811333326694300672662121216288594747020130621825297208162301715421731123030040765
1/5
1/8:
0.1246103484114850115848525439312499915920286655302867263274069401503016603060757969577410635770990479
1/5
1/6:
0.1657448324175337912095972644556932261526863654827292929554186166571911737385310476208762522368234431
1/5
1/4:
0.2469058503630388896536133804392056872707047070055011822481275095028371802644109838771404675445432028
1/5
1/3:
0.3260546474403722702904950285484049636583547759022462004444970058157359435826827569379620329824555239
1/5
1/2:
0.4759506549710993625260255025476085651414554651214904804604983864089098781997070189120656848517571059
1/5
2/3:
0.6112648179985132303660538522447882522735259796961937332457208707498012519071483247679695492795995518
1/5
3/4:
0.6724226282964769210393190772403061973294289492092857341200880145806971277548993474090345292935175088
1/5
1:
0.8264684982946884270997228340503759457810085351063667260969353522655544694642182185150090275833073244
1/5
5/4:
0.9329132654292365786402188592981148046037845139086823726913684069315768454659328605398100118168270804
1/5
4/3:
0.9574215287968981179981976699552101348021415582823707826550631551685264178973949081280418133517538969
1/5
3/2:
0.9898082283807197813410268796546764339837457642939957405114629822955888647919037966043970693498002073
1/5
5/3:
0.9999801580769434208236594581795988172841477243865198592299931117398449031245016847207967530834513302
1/5
7/4:
0.9967328459141865590420959387748025413076650895695789086603101337760500260302404564744440924335639273
1/5
2:
0.9536674309961750776129662720688869409265816207492193457918347454611804998255257679006271520897394483
1/5
9/4:
0.8608461972355058344593219167181833770363550959052407983847668519135258277409927939357439075152199922
1/5
7/3:
0.8190239064032670087957875983840348240202851339097503391841811484870330315521485067482078309926133602
1/5
5/2:
0.7196372132509738484471947034160011886524769970711637412314067733241126753282374900128836312729201164
1/5
8/3:
0.6004778733799406278961768761762788318228434955347267539880576526213073899959237180966458033251026724
1/5
11/4:
0.5341472367164387671423992279661555944901263989761492104688121961114965082195321272645474010159044410
1/5
3:
0.3128436617625313798875448133919282397459617272034910601750575699402341986097034416353994216270783444
1/4
1/8:
0.1245942551883152381508079584835281889328581825576187214610501258910367400212337092060614822889852342
1/4
1/6:
0.1657070195648379202019358836204264079408865690878656775267988763376381020298759657751720471194833448
1/4
1/4:
0.2467814051361416004836237411012553897438474130138171886327388535461841670389316231510632663486540619
1/4
1/3:
0.3257699203040997558481393531239923213426624603259410848551105107190047135028053775265600813125315688
1/4
1/2:
0.4750829360285365100822183247038702587450781718074289480282519303601255717727379759089286555996797550
1/4
2/3:
0.6094872570738714000935355307112575636112330807811404530480232789276423388221683161717858174234426421
1/4
3/4:
0.6701113282238753299945023434033704669689360027679251342948442365373321718920440749113511328750413168
1/4
1:
0.8226355781298623596762303386539764884406471165105568454715304762810372716303433965933562306145741380
1/4
5/4:
0.9285612364263197757002043882699991307827119022034152393995787949153247910621564269544265070417867551
1/4
4/3:
0.9533186077729029719749605075690392140950563900262534936482850711894367792290187100970839935077586684
1/4
3/2:
0.9870522873112416666485891645648094372479695625977041645758671802464974970729363006422573775743589982
1/4
5/3:
0.9998634287816776158006894683389943011481134912977803114790948479469996387676989945563694820881220750
1/4
7/4:
0.9984518610208355964665173067263647027764303953610669578426602874690896446733483197024578612897588259
1/4
2:
0.9628981775982774425751398921374390419780511842177362982619232575167653759426435759559935520800282252
1/4
9/4:
0.8800380433085542144622046013519134073405452544251200755273466855202888612397136097397919733143866626
1/4
7/3:
0.8418683766214577605539303880611392627464487284640516152601265326501711299509211590711157230833358594
1/4
5/2:
0.7499030499017841436699449976303955146582200167537056054418028625419420344844763863511097505036119202
1/4
8/3:
0.6378297315229391446480842518197487317009135893225327414632654284965978787701688195139926518105426880
1/4
11/4:
0.5747297495495692623829249982593503360857155323857043106765715783958257129976497566722421888457655709
1/4
3:
0.3610799872648924676555523594682352877160964376103416900585405060898329646313330864908455664341302664
1/3
1/8:
0.1245674358341209106248755253834486471100666610422910891846561484406521542366954834951865046415570270
1/3
1/6:
0.1656440090011642447790493385830350751263878950008366605038260623912015851775949634531876777872913962
1/3
1/4:
0.2465740693092298326673196278613468465055468514205932512940275866889030942271218192276738613669900763
1/3
1/3:
0.3252956288061814149207558924978986894132234303367894555157047430932404044957144906485037343409800579
1/3
1/2:
0.4736376403693559018271837345769765773713852652185150695037569585343625215192982788083678415680823236
1/3
2/3:
0.6065236256242167909489289111671653918499958287158261746217073240490798523372791294209085673322708737
1/3
3/4:
0.6662530500065658197422415123048354047208443070284008308738141821554101159953686478030994540823683385
1/3
1:
0.8161766374798108428297875974448488749198594141266916067913213308727005675353142838653698730736290066
1/3
5/4:
0.9210159521157253486078849378093940124953258725682567681320224514827078369244988918547747850709450108
1/3
4/3:
0.9460630398845852003739326012029810344390547873701984054961567339744916325785805171055306114760542767
1/3
3/2:
0.9817066666055167518466811997480929861017079165921957639156607187077212566246275323191977096281983246
1/3
5/3:
0.9984920912388914568659218655532299498837438796241908462725322365516428501590335264100724084490590521
1/3
7/4:
0.9999137765678078333297746855333096894565935677056209740153763799487179032992121918611606953914391635
1/3
2:
0.9763095827654807192795048736091604388218734005381000372583841746013372334122407557632344511884807855
1/3
9/4:
0.9100381508239219113295438312104881335598337894611879794999341578665410906081785158351549625062112643
1/3
7/3:
0.8781676850635639789549476235111817016753234108038740775175450102108534422292504799307418674633496252
1/3
5/2:
0.7994083068523778927282044723378430090220672178931862029950908625851814488645812685433183817882506734
1/3
8/3:
0.7005626421884253930149520467114540519455553347272692861517883777243382125623296657779204863966521754
1/3
11/4:
0.6437556973555593927066558898242632603685308005441554492847665241245463727147299298684755099332278405
1/3
3:
0.4461021659191626749250494917497797054361086007485417627803572830904177707208160987239150512049924154
2/5
1/8:
0.1245459827666054347101496990941353988955169854994162743162934073581144729949103165897410745536709231
2/5
1/6:
0.1655936103211932227941960700637538093998583839013883632532790299983988411293079559914879807904538744
2/5
1/4:
0.2464082662433973903937818883543262356297906721405595247945717035328358923639796402029413797317702975
2/5
1/3:
0.3249164240798080916888664549839540181904613164811064744261280555712954116636696197049341997017281391
2/5
1/2:
0.4724822225641003928692325463829128183991326726111001823111222238500844008959435534934241388385984592
2/5
2/3:
0.6041518679219105871643347131832508091443919382294185561297492044244529031896438672426443911525642947
2/5
3/4:
0.6631611143251621935550589294505783781528622332183696773652502882501305369562050409621761404103203067
2/5
1:
0.8109471479042083557975766502113809139387677208020725891349441716515093828635987784691544239238692843
2/5
5/4:
0.9147202292114018754065444083121364332057865409036761171488213586655598158386286947634535867574202941
2/5
4/3:
0.9398858533539616553621637299421709059945233691233964246204223066895515912613768012870829084516083264
2/5
3/2:
0.9767495302544841594683587871916698321246298708487405061703548268882077516569290061029884354409414177
2/5
5/3:
0.9963097401006033177462907590290267958223933007043672241491550369897868564921333872022601741625828867
2/5
7/4:
0.9997727909224025473175383706730787448238119709465631348389878607585922654614826323867520716925756378
2/5
2:
0.9850902348663265016931267540794737270196678785860436647259470243949566778173092640005087602659590736
2/5
9/4:
0.9318544093987840530365717059108218506665663039236631763744593045474673909374843173671329568025451351
2/5
7/3:
0.9051260100085541359066752306443931728475701596540243806385386690965379963163932570936849739463345343
2/5
5/2:
0.8375224651285249761784836525158009915998326993358332809577721105579361475053558696912580567605273585
2/5
8/3:
0.7504464809567373864894198266960282155999002148268797973184678189130256381926888056536659690286018456
2/5
11/4:
0.6994991186987548048401269197026772859110511327424859571770941570049138033297824277491434032817220386
2/5
3:
0.5178801915872889976029657113730552632579233477340195241072047758986559621493110860160946059441110426
1/2
1/8:
0.1245138071914409778458438526615376839031257970873701282130028925409009065843171844551602459562195310
1/2
1/6:
0.1655180285845793839928080652058602196282597155400848323342896525197485186012627184657872579099659919
1/2
1/4:
0.2461596709698614583325382694251354344648942818129828367001742732325646054456953138490626141372393130
1/2
1/3:
0.3243479980165896578065521474936206218109946310638477977743294778395950023172029193359253913413645664
1/2
1/2:
0.4707504736556572833323918882921811847399595331739569598920121989833659673392060272024652384312888340
1/2
2/3:
0.6005929646029966586745224134081343582988704935166958831750689695543070933741776538551855967544426550
1/2
3/4:
0.6585147441299966807918415147240154341731484968329167465224735185749618539838070589249547522456524865
1/2
1:
0.8030018248956438876393973428189896311932943191842263831897953684059200200939163770304934479798634673
1/2
5/4:
0.9048515139796526227334736813913555914955595105418707492656625869221954644576809436952055555019600784
1/2
4/3:
0.9300061909935596910393790402451068823396784413384348457133518952533408513516981409729441417477374558
1/2
3/2:
0.9681760156756912156216033350364576279326652961113802618218355918145596432440757234032124565437865030
1/2
5/3:
0.9911812154350618730115790729882941962245359712317709076490926094657043656716291156327764160778851525
1/2
7/4:
0.9972884550327895410909190845928004514567874440679269088215107704780042097634899005925057525290889505
1/2
2:
0.9946623253580176836012543020534988301654788124144726070592909154305715022106622716783089015299228890
1/2
9/4:
0.9600613465491084268485567052520920551859943442178412993846601045900917852253921677530635218249173906
1/2
7/3:
0.9409877685675867711917881738219590797885741713329253211527585673137247129436009662159677492676641605
1/2
5/2:
0.8906151882260943559473944250688473574877589893439795195838957882761436479066685624095698297625403540
1/2
8/3:
0.8227643769635372247015713109827389437193409266895892065971871071437133218719828184124608681819165378
1/2
11/4:
0.7818666657775244449550595482563119678914882311057797259263536218921131652690077574714564479511729826
1/2
3:
0.6300289982420331644946371391201116394839643987284230472988063036681007762821044572522291942236431025
3/5
1/8:
0.1244816364472342485765218662074615869707036812334459521980143441805498849618480008144023920477970596
3/5
1/6:
0.1654424663859230556946248136233860113693937552443248725416912654458139107986799312415148825021043224
3/5
1/4:
0.2459112068852820452854219268535062592281052485607391303645658869811130294890730703250245936612219742
3/5
1/3:
0.3237800294899710645669393392903125585204315527089437883790027851903113340756643522099265904644928522
3/5
1/2:
0.4690203949724963526795208547389784952230222571232876133700925387746939176092050243193674666405116183
3/5
2/3:
0.5970327372392227832498676780686765758582675309955204768424176739398660977187305038748963380692529344
3/5
3/4:
0.6538587089550234503266767174862618864277736809205044767507863792901059196667535376697075136679641731
3/5
1:
0.7949388393365780942815267645969199424526536324767516927772575000135158979061162525482530450755316173
3/5
5/4:
0.8944830247857194759519407177189265353163105006133990475268767961927469588907151787826620858946683762
3/5
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0.9194011845152527690156101208730807533990510396406062884822534617925054022560344335172429236767963797
3/5
3/2:
0.9582387047365207351491301265411197709546998596176732779782655936891188887466245853082370859062891980
3/5
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0.9837812391956268215108618590089287730184142171180748709436121472851295399192286584264569660271502531
3/5
7/4:
0.9919818336919809967350402010599878104151118274685563543294425306066755688426648486897090424818909539
3/5
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0.9994911020390567646797532054777845583544745711267659091661250652540662106145420631281588441378373335
3/5
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0.9816784800612060156642455101227157985928218953048434032903660236285158362808909874217848871357450869
3/5
7/3:
0.9698303858668241093325086997571854822176065281753691138931241353617811185546241415329710226117093547
3/5
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0.9364232076981755659442509936566627653009958077152924659684445443169560676158665586436997584721281515
3/5
8/3:
0.8887428046470611072821475840663244533639650913769523500481167082048933663619660983827199880002706003
3/5
11/4:
0.8589830781177301092472658142944748013475966237540959150727200463404590994219845365187775775324991462
3/5
3:
0.7432676860864044186189152749271150935118937174913232896201934808965388443482314263018394212901187577
2/3
1/8:
0.1244601919680880558960572455119962241150973443026275108514480641717833631940287153975266554590057348
2/3
1/6:
0.1653921024400982898725987492185015870822831478900412476356823157625446892697894979880490865163111673
2/3
1/4:
0.2457456370442167164267097254282320294379357276382235415148575100189962735940076456461074892313019495
2/3
1/3:
0.3234016381610066041079395745195708114474858887043618657128395045797650225532521971437746038210629633
2/3
1/2:
0.4678679463983641430276313552508480254743254455369082771259933503701523513314342948451060795116483152
2/3
2/3:
0.5946586250234656882672619285560219657838858411601763906557789021787987622093258018186825117530414365
2/3
3/4:
0.6507495898404787199448287342636543934932933184229371638504340813663329121332953350658918252587200940
2/3
1:
0.7895001970464455396373266566913564587613263592724496672339530965217506031765912705514572274416172028
2/3
5/4:
0.8872993073411089579192193278186065709192622281280298079141681730528456879564999314801122512673967269
2/3
4/3:
0.9119354974679931468305391893923898424343641762684384373333127184965936907256530104340848024894501798
2/3
3/2:
0.9508602509944422046341980873526735439494359794081517864111673361997474797989663582908931246041410220
2/3
5/3:
0.9775686006911813545145330482693015125725735180661757422265576224889929181523040574982714956285932854
2/3
7/4:
0.9868346382417523800450597703701183715181506787979410330798007650552561789843982822204681294906596090
2/3
2:
0.9998602056032434982443899328630548216722750689703457844824434069901999319783737268077335045176911482
2/3
9/4:
0.9917795990668127931222514148961764927723199504053352905050630325559217778945219735725916261154669504
2/3
7/3:
0.9842822331973577995565455797229845464682803329563256881609187368568656690404148185146924854761619481
2/3
5/2:
0.9614392936188470913531265440273752989907032148249863133280758032486061101214808072695485450180242153
2/3
8/3:
0.9269609637061431330369526006289189223992169101596903487092036948198975862414581854483156284837799614
2/3
11/4:
0.9048086058249164700040862216999755659764418736363596451924873441383863372770418587965802785044553952
2/3
3:
0.8155825557073725014271165480604845814722150671326089898820596536777111312456311340841323819995287129
3/4
1/8:
0.1244333893880524467839281141813526872773199683546539017945212383716036886611200105837846360479084236
3/4
1/6:
0.1653291597165524367537892395873162807175938920023847609228337962692946499241122463567918623049057624
3/4
1/4:
0.2455387567342640381712588914020381747894307656200541238421983661717984802119271012639135418318830498
3/4
1/3:
0.3229289353262448039893364992440085229598761413133332728624487126649170601847771152596874619411451054
3/4
1/2:
0.4664284498036358501604956225537041712797924723811324349181059887161547723464037734065519842060938440
3/4
2/3:
0.5916903924936890035632429171047353913596492194784281811527760175687232068536435283211619287133761627
3/4
3/4:
0.6468577456289555559968719729012195112196769238733708986643322986069474620370094741661006098408502198
3/4
1:
0.7826328478392893284752533829097309564479277774855887301413042688146025648117082652462565118577839573
3/4
5/4:
0.8780214745055230318349416922448411400431282464335738339113190171097273074175766027601106725691566972
3/4
4/3:
0.9021670126295022410597877174213381175737738706693034751052867538251130111997722026046585820119572873
3/4
3/2:
0.9407973092127776445772602581525893891347651425372611813447740118574167671362693002570176256059691581
3/4
5/3:
0.9683544760387640143092005566923025754308386167959550451666156737103651225599204865711571137162195823
3/4
7/4:
0.9785599447422769131086780362043304950618265635223154893459713613100256946970898337580077894761293678
3/4
2:
0.9969206618889345509390618771968539288622631188736319393066598646111143440188781089429361403780019961
3/4
9/4:
0.9989053964463055149946268183404669849977106945488457312482580919594548007761804507199771671249884557
3/4
7/3:
0.9960639297110218667815668708121997389716585652338812138702124012525212470293701850965963167081770350
3/4
5/2:
0.9848532091344996552602080873711273067332541751985660860470930686115348787747941654185603602452158231
3/4
8/3:
0.9655254039083336777800679047618177350558451450202691231330071452162307747641967308946550980401779396
3/4
11/4:
0.9524138970082537028208977088934644020421394820872128805063738666888937230046259745639900093254529510
3/4
3:
0.8966015302833202990834745272812004102248582145298446793066834422043790047936650968250020066982088986
4/5
1/8:
0.1244173094500163427552226187764143076658459630732553956755652224935471467355965651391977934750590963
4/5
1/6:
0.1652914005932363139149302886377201818016648585587053430994892136309144923442363611868551917588653016
4/5
1/4:
0.2454146722766442386792066504532517961604569008446342281862408619140786508902160681391943407671016303
4/5
1/3:
0.3226454664077589611834508398028300910361367386974019339220360946188137557450213654839568386680092623
4/5
1/2:
0.4655653235182289746858612635588656740057919529345224151121825013508698176226278061765406649168561997
4/5
2/3:
0.5899091852413268144430760184089916531841078143665742477328063440435056645041318307742769021310006239
4/5
3/4:
0.6445198569711695509122269620800313211091204233000978014281272787682681496435216239324668336940478408
4/5
1:
0.7784765552276773412676842639271891610728163342958956092783062170540214160825600820489694515856703837
4/5
5/4:
0.8722989688507077088486365677417151457668978136533432561389579627486500898814309252984164970062041617
4/5
4/3:
0.8960773134094114060214358643969204772842296135843540154666225885579535656811113611905105872679444681
4/5
3/2:
0.9343163001725960172654869323050935269302795385129242717324347741554548765359265269744299605456086247
4/5
5/3:
0.9620532309182409844698086361640500799456672512044288948276397170734875461417221752237570284762580023
4/5
7/4:
0.9726060069160912915531979244351728094639683135191854280948988290303065721868219451612791872839686417
4/5
2:
0.9932748774948721810103744316610429327951371224492731621044318005665389340902393104636029396346054985
4/5
9/4:
0.9999948082591566594822691755673258798353751972210189260677338975677952324392147462876263880183271272
4/5
7/3:
0.9994196127998595637187556432982667906931153320176677867898230491842485119578794325603792193312864971
4/5
5/2:
0.9940180192187143551560630782725196598133476753731197211895559576271164662489495060643719108448364483
4/5
8/3:
0.9825274479259607560262701091931182904513124278650914366046743770415628877890835750761621700631000903
4/5
11/4:
0.9742293476176095952160921161851827701554485275272275084942747766229193550052892998827158689835719778
4/5
3:
0.9370985343512131348892679535694623693627322404270068446241040760514593374686807551591384459506236102
5/6
1/8:
0.1244065901621204337077586599764133955030230184666410981841424221355462262869127494058247915158905663
5/6
1/6:
0.1652662305570184584611042589908124720635856955446980080584095474611553062358538691888188632579347501
5/6
1/4:
0.2453319675249062351534294982508748689754620378479375612373359638587043829033943562178658878636620361
5/6
1/3:
0.3224565508140008940268200064798278532883819622297693958017104925654408178238655616802409782190570057
5/6
1/2:
0.4649901456219161231461556101393310231473712718309288954705650498011512463927161913011255477642595867
5/6
2/3:
0.5887216188240155426886458458702264954896439056378303892209302580702977622420559919740279685628070593
5/6
3/4:
0.6429601475348803981517835687864808502960539821173411389090661354935442360699975920346426819774629623
5/6
1:
0.7756910712576395352626136228022730247943509889104607599274604835403026178204758173872597782833030151
5/6
5/4:
0.8684202143554381363513553249048879873820044544401439363280635201456027732338066722768222859871474633
5/6
4/3:
0.8919237940614775915519739116277782514613854295696543536294778909243476717019378023580354977762040110
5/6
3/2:
0.9298129246613924411874309030034790948475227394541338788327139480887143570339300416940440697618055135
5/6
5/3:
0.9575312839809490814486486462908887215927564646602825770264717373667095705599174518307938951101618745
5/6
7/4:
0.9682235421906839299714166136415480825330805049772474158818251262362765504775648974868795215519340285
5/6
2:
0.9900411175720416066504410325087183604286936768828534737327661158321177969486449421303871115439007572
5/6
9/4:
0.9993160477920040361385239845833250403644808411220211941324778009608839896570780992351700576606920579
5/6
7/3:
0.9999957233502503686701039197480192465624126668347087586535920107764704865738975738918826253906589967
5/6
5/2:
0.9978656485829333532647024702309135141535079427167478198304857562688444111870810945216004404966508867
5/6
8/3:
0.9908841862382444061055528461100173436482449554231753088421627178739626967126407477443661056385410470
5/6
11/4:
0.9854019245011042525377693146778677107488698095085429230209370541144839733698917918374841097776762719
5/6
3:
0.9595080880507102130386688051229322768967382536059225884304425430535442278011510051286496855030253508
7/8
1/8:
0.1243931918068674186811043519963731585062898990738472778707995232218708913531734425416411653949503903
7/8
1/6:
0.1652347710633374832379488466160900172529638770517090807808726231119998769193599984413940365246413906
7/8
1/4:
0.2452286070826231942311898665468916745081011307891092456180849423938872633785357180742152472532063917
7/8
1/3:
0.3222204779894712908162511906393026640198998951493893927501822962144521558911964145331771946499759168
7/8
1/2:
0.4642714440347310128301311331582455003496029624505430464788845265003481224049457413461445906756083744
7/8
2/3:
0.5872370680061799473342153063766057903161650463577763123330300178685455824743464081470217170575581272
7/8
3/4:
0.6410092896265825330724270108305300335978456486363302813212524000331860704319964666661494530499523444
7/8
1:
0.7721930453967359304445234790115858856130639693999806199618840113652876634820594801339657787993137707
7/8
5/4:
0.8635010725361075373704346017585580241495113475116601084329127059180065980562812135132181574719403570
7/8
4/3:
0.8866278281838486988755320615431872770021999209447800673161959349516318509707889299316098614332264123
7/8
3/2:
0.9239801013259695915158807030776284704515744688144611502545679063181732435377715202259263681571547542
7/8
5/3:
0.9515190818611556847466439203405081035209668007954686965665856305933878449983955045117777034566966309
7/8
7/4:
0.9622809583603160983121729349094753743065825968535466276152504685223268238910116985861270422158469513
7/8
2:
0.9850830752271641786562000801738228474785483879696113961794764349842625199892035260226268571644861492
7/8
9/4:
0.9968311990451370911283387814457302202504350558235751071306083763574383511713555050235839763202274045
7/8
7/3:
0.9987635824726469947844441974680340881375115362730579384993875706566438184745613316520436668078895653
7/8
5/2:
0.9999564188284435885686663777009223504153990368182950497953889823337011600203941109532588653414746443
7/8
8/3:
0.9976457446713681732766218199611722809864011291220487654323760878283770235809639678856651695107699165
7/8
11/4:
0.9951218846420401772063041859596036232750519081553317212292785029045465191609430362549730030947404709
7/8
3:
0.9813046769782764525414479699759676074842638007464801965978431295417664961216546364258297358679239017
Definition
Jacobi's elliptic sine $\operatorname{sn}(u\mid m)$ is defined by $\operatorname{sn}(u\mid m)=\sin\varphi$ where $u=F(\varphi,m)=\int_0^\varphi(1-m\sin^2\theta)^{-1/2}\,d\theta$. This table gives its real values for real $u$ and $0<m<1$ [1] [4].
Parameters
$m$
—   elliptic parameter ($0<m<1$)
$u$
—   argument ($u>0$)
Formulas
(1)
With the quarter period $K=K(m)$, the complete elliptic integral of the first kind, $K'=K(1-m)$, $q=\exp(-\pi K'/K)$ and $\zeta=\pi u/(2K)$, $\operatorname{sn}(u\mid m)= \frac{\theta_3(0,q)}{\theta_2(0,q)} \frac{\theta_1(\zeta,q)}{\theta_4(\zeta,q)}$ [1].
(2)
$\operatorname{sn}(-u\mid m)=-\operatorname{sn}(u\mid m)$, $\operatorname{sn}(u+4K(m)\mid m)=\operatorname{sn}(u\mid m)$ and $\operatorname{sn}(2K(m)-u\mid m)=\operatorname{sn}(u\mid m)$.
(3)
If $\operatorname{cn}$ and $\operatorname{dn}$ are the companion Jacobi elliptic functions, then $\operatorname{sn}(u\mid m)^2+\operatorname{cn}(u\mid m)^2=1$ and $\operatorname{dn}(u\mid m)^2+ m\operatorname{sn}(u\mid m)^2=1$ [2].
(4)
For real $u$ and $0<m<1$, $(\partial\operatorname{sn}(u\mid m)/\partial u)^2= (1-\operatorname{sn}(u\mid m)^2)(1-m\operatorname{sn}(u\mid m)^2)$ [3].
Comments
(5)
The parameter $m$ is the square $k^2$ of the elliptic modulus. Thus $\operatorname{sn}(u,k)=\operatorname{sn}(u\mid k^2)$.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.complex_arb import ComplexBallField

def jacobi_sn(u, m, digits=100):
    field = ComplexBallField(numberdb.bits(digits, losing=64))
    u = field(QQ(u))
    m = field(QQ(m))
    K = m.elliptic_k()
    Kp = (1 - m).elliptic_k()
    tau = field.gen(0) * Kp / K
    # Sage's ComplexBall.jacobi_theta uses theta(pi*z, tau), so the
    # DLMF theta argument zeta = pi*u/(2*K) is passed as u/(2*K).
    theta_arg = u / (2 * K)
    theta1, theta2, theta3, theta4 = theta_arg.jacobi_theta(tau)
    _, zero_theta2, zero_theta3, _ = field(0).jacobi_theta(tau)
    value = zero_theta3 / zero_theta2 * theta1 / theta4
    return value.real()

jacobi_sn('1/2', '1/2')
Links
Similar tables
Complete elliptic integral of the first kind $K(m)$ —   holds the quarter period $K(m)$, which sets the periods in (2)
Values of the incomplete elliptic integral of the first kind $F(\varphi,m)$ —   $F(\varphi,m)$ is the inverse relation $u=F(\varphi,m)$ with $\sin\varphi=\operatorname{sn}(u\mid m)$
Complete elliptic integral of the second kind $E(m)$ —   $E(m)=\int_0^{K(m)}\bigl(1-m\operatorname{sn}(t\mid m)^2\bigr)\,dt$
Complete elliptic integral of the third kind $\Pi(n,m)$ —   $\Pi(n,m)=\int_0^{K(m)}\frac{dt}{1-n\operatorname{sn}(t\mid m)^2}$
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds $m\in\{1/8,1/6,1/5,1/4,1/3,2/5,1/2,3/5,2/3,3/4,4/5,5/6,7/8\}$ and every rational $u=a/b$ in lowest terms with $0<u\leq3$ and $1\leq b\leq4$, together with $u=1/6,1/8$ (260 entries). The moduli form complementary pairs $m,1-m$ around $m=1/2$; the arguments are simple rationals and extend beyond the first quarter period for each selected modulus. These are not arguments proportional to $K(m)$, and no completeness over all rational pairs is claimed.)
How they were obtained:

Each value is computed as a Sage complex ball with arb at 64 guard bits beyond the requested hundred digits, using the theta-function formula (1); the imaginary part is checked to contain zero, and the real part is stored.

more

The generator was checked against mpmath's Jacobi elliptic function, against the identities in (2) and (3), and by inverting selected entries through the incomplete elliptic integral of the first kind.



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