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Stable Faltings heights of CM elliptic curves

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Numbers
$\Delta$ 
$h_{\mathrm{st}}(\Delta)$
-3:
-1.321117428428037914989869195854823463769869753613337835876975111374731300511217673285519411607513087
comment: The maximal order in $\mathbb Q(\sqrt{-3})$ has singular modulus $j=0$. Its height equals the stable height of Cremona 27a1.
-4:
-1.310532925911509518252275083304728667951607589407810061122482499838460018439289916934472670701064191
comment: The Gaussian maximal order has singular modulus $j=1728$. Its height equals the stable height of Cremona 32a2.
-7:
-1.285615926319450743400649895587228947674420151850350279825209410256368687488053877373367158638962969
-8:
-1.270406626059825756527817757437320877343853973484838073206274904257889630679710858102796380185310142
-11:
-1.222546642529067085686657254629906320677012025125121085947630637384629513615993691452001285766444528
-12:
-1.205592898334713696753663842278460702423953064553295293523528443125832363516268554017875523774776639
-15:
-1.234583723140373246972817398718655567987540927080577794789062875936832176826886035992564166806652735
-16:
-1.137246130771523190897967052940184525932732555817746247592312497465111612946866238033006838951959519
-19:
-1.086258404682384938406532296064420721350664247563655558218714434719808582668071856548552819913745435
-20:
-1.189735463071988102094130362869775620008164589946160864322595430689955667445974292149123512577597199
-23:
-1.207098228617075499209467945766842877115356482675737021188021790384500169059012506961027384917288866
comment: This maximal order has class number $3$, so its conjugate singular moduli are nonrational and share this stable height.
-24:
-1.150162287412036732007601914733732320627402676950003439155266207632673437499042520894596436495888932
-27:
-0.9549133322053346845247874502139815622207062343390880186320770001622332027716813509943141600029357242
-28:
-0.9390423360394780886920338348581406636366700846702226527648694055096718765032065195704354951407536249
comment: The order has field discriminant $D=-7$ and conductor $f=2$. The prime $2$ splits, so $c_2=0$ in (1); this is [1], Example 2.1.
-31:
-1.146993083519722972212834578285679615246486754102925580453371138950829994241016519054966155562201048
-32:
-1.097119830919839429173509727072776735324978939894774259676104901884541225187287179201330548436205470
-35:
-1.071920355599063113433758947553306862081828005569686837836982012996028018440120954262693040563507391
-36:
-1.035879853744482095403463774074097241789734949952122698188808916429086445134637675216068731997631169
-39:
-1.143588378596267734466915470439035025029790133527411284476204246451056610800683038275886484031402237
-40:
-0.9983645365231737967640856838449976207666614809701049953920362407043739332094234038406463541682591499
-43:
-0.7148701161205011970217995060000093609673379490612559478594203185028748528752450199501190382684021658
-44:
-1.107022112435742867450451901053543559331095336065078543594183969135730576621044572184357397933708080
-47:
-1.137953967829499568179572100372299482372935048236613857504643449808909724177866891169405674532001329
-48:
-0.9745438381480652602812531351257351797321196864332102088166351066280344895263703154825877481093037434
-51:
-0.9537570519442836366308768096926935334044319526809791503489126647506947059602453279368077311837890367
-52:
-0.8899429766166946107880407556949550441142908926791282769480029975188349723674528685926371998409172371
-55:
-1.055482114261673717185445486393779755201268324886545007131336658025852967469557140497157638308649419
-56:
-1.075344728622050699006266036728566816369963483049295067706516926844412905535335719490112213144602156
-59:
-1.027092396543272299966795150516279427872382000011415226633394742926217440825515927635055475054175411
-60:
-0.8880101328604005922642013379895672839497908599004501677287228711901353658420386781896325033084433912
-63:
-1.010962854152423320551838586356597521512547512394662916891535826846995114183401635654963219935529946
-64:
-0.8773159380615436998665050073933683129044200054326505272970574939050890047082307196808080913283025113
-67:
-0.3995562863944231894278294562794520030349474404272338586058746073168554222038544872698292228306822501
-68:
-1.015595251570772041682897450578875544643582967795408761699904405127000646360924188884438708158719916
-71:
-1.105528969792081296398850785499625900037768558868258967699520316565106700195889005959075221332542640
-72:
-0.7211004817257709108301951389760580250201086945734633473389277374391424840704063746659885027784440980
-75:
-0.7846381242833377901229494181127609172613359688571652619060924808833383046086650850754747855764532169
-76:
-0.9707338745890607201703269424880579600047475585036130158652677664709096456731227372809089320810089876
-79:
-1.003789349898613685175141137511544763702914702987745137864994647960470466668263899684353088105078989
-80:
-1.016448667932001774739822332505231477989289556356097050792425428316607261953550613247657680828492527
-83:
-0.9047027607680709741012458609941213355606561010112958907102553568698294837531292593575989927203597296
-84:
-0.9553876470030609416809012836829374309860148936782200370068504039845338240114183673044594155280379158
-87:
-1.016864903807843937389495703663722747632016884624314625974444277387235762015799675240652309995586491
-88:
-0.5954285243678658445727593399580505035403566469475084755566384082038969560786948410852983073996545809
-91:
-0.7189929721166742820172971144804180169360834556338771919612321745431953598777943471752737733840440934
-92:
-0.8605246383371028445008518850377545930776064154956093941276817856378033580741651491580957214190795225
-95:
-1.058452565559257702682813832235150792600598658668879792059026536307298497176883347605149633167886306
-96:
-0.9768754922720504046532938843691881786085276433599396256250962052593250320066188419931306047467842604
-99:
-0.6732404981950122399890346361686434683532667462137463600802834705658823670066892080151934083595784835
-100:
-0.5058139696944593309518954166916348481888069122735512001661585541013705245854610346194057316544743856
-103:
-0.9058059811186217504836033963208418317664615255191567374900428297353311401866278218727397068569433881
-104:
-1.002094999219886898584531588597153526983976395009431602105919703605809292723536058913064708766356651
-107:
-0.7888079077435625421862400366648543973174457641395925874455846385617159546615083601748738929835747052
-108:
-0.8393888021120104662885820966376188008747895452790454762786303319133342657767322317266702721701992763
-111:
-1.024250210310935798441332032103179694310332680923348356423467568734489026682687289015860895706727339
-112:
-0.5924687457595054339834177741290523795989200174900950257045294007629750655183591617675038316425442812
-115:
-0.5918685847632052678584447114850867628043946208687686622273985238714219342075265017739394059198283436
-116:
-0.9641749311381702285609617652003241042313690144442018556227298025193236220158292312973182227689584245
-119:
-1.051827785830202368130572795626384937848258869889554850470547209107479492329941702929621855605228488
-120:
-0.8046749229637282152767145161867273401547811903073209677889772766200826185019151913361043304462170832
-123:
-0.4840417530014056977372705520370125189382066048824387025458035645934280841834403990180770000337731396
-124:
-0.8004194932397503175042185175565913312087366869227979533930311342041331832561691612520344920639917038
-127:
-0.8112162781592321449480948821890724091089961885296415871774768877715909300549001011335548546863939760
-128:
-0.8371896382098599381420476815259605222966663895096785393808498983245186169486516608491318008125484626
-131:
-0.9301521843477471522215425967104891464602885981470710233948263652055799175001521699869491686001526153
-132:
-0.7613537590259701774334994352149926937901072497786245787173436682211536147976618833557351460644391822
-135:
-0.8683796269176700165077356530778136664383774078063279775441647647243340790873497137013589152020753721
-136:
-0.7800490736273831752765446103735908536597883729570115129680348473086187245351936561749419521277020144
-139:
-0.7473891885142228596657267314370087347534310766390124963788323114288403044530872373885721574834301216
-140:
-0.9563958255057388951975535939769441007359113165096442954835353447471290814451718349950491527307709428
-143:
-0.9978393482954283174888038874258131665269750523730558317243389554799546929024232025750790756480674660
-144:
-0.8625930586044957680491557437095530997708599163620588846586389140557380396422139963146029002485264969
-147:
-0.3481623539003812624371928241332335989513273888224072416472800364059413691351830393916901623135773237
-148:
-0.1786244946122713559175282021910156449143361411072358413373978577291061800287786690945146632098110364
-151:
-0.8859694245775282151610118239678714859165686793928264156169921297133907301907267065277635025432507004
-152:
-0.8634422844579651636753206934804444516167435605604900059555996011742009183228137496153255424814111181
-155:
-0.7554838674722786252555603654193580643195568235807225815938265786954500889063228543131158279669240473
-156:
-0.7970147883162950797582994097099467409920400663472836574158642417043597998158356804729548205331928935
-159:
-0.9779896561761484348144661966487386878855377708151675283005878913781097244349521703667593404366787708
-160:
-0.8250777413831874694097776534804534787477864473800411818618662383310255277169997249391805224191544781
-163:
0.5776898062633132262929352576143725733879055677495641677876110456775568527548555909308944205058160072
-164:
-0.9449543906810882534780285376365628647973257633616201697854042987062217638867929794053776546026092251
-167:
-0.9771308419451690490647928553458571352770843651334380751595179515743028797994161860708122670059946848
-168:
-0.6271346707999431153769236516294677015128022912479711186446514212307756770742575807284374655573640691
-171:
-0.8116053325153575155577209868337892951887916081079681952850408513104350093634196148301488812103124133
-172:
-0.5993455860271769787855941524236465996214212600012134055059736502539759158802959006824751504356657178
-175:
-0.7491366221747506185337301178451664011658863670941777058543267797649756915855012891633225326079030984
-176:
-0.8759730522490944309780411939008180366392619579449934588872906326379327026311463336490696222682351841
-179:
-0.7986382570170114859235306349638919774338290860177810969850186602254069259391304605337378347654763194
-180:
-0.6404293187379332563965077444085127676844193110347861384552482638712085208366698087123156351707311549
-183:
-0.8392548142420279570476999035427343974157370645274442025851234842286067509740845050629831914306085029
-184:
-0.6193660539476399530126345220157751565628116584055981340830658652152218428388502086373950301737471777
-187:
-0.2663841197920625207485183288059644974544377677144399693719502216186448282342542631645522828097625235
-188:
-0.7913803775495269134709560396432111983351849810564862304443034450622129131930195333664740110337919853
-191:
-0.9822448597927468476537059493875948685978855917058288332313532458269958142514829870867333528459382119
-192:
-0.6857325129147547146907397511848282763673279637831038529330184360057871470389975173134780285274626236
-195:
-0.6950626850550107107687794801212759800731318342271881584068495480440480763163321260622378121632017418
-196:
-0.5808166200157670288377678045135362693377008158146121154502111936118675699072639415141007337306123684
-199:
-0.8817371814736598256853196171733394302336369536361413777675651944645169767865799285266025139245661591
-200:
-0.7339273219151256316608979796952583308353201887286654992353922737664966347771582698927517541542502721
-203:
-0.6151350051754636488676896353308856699934407809660054761604027961437359814121009327788732553668112237
-204:
-0.8382325218509594183946714561163307720585152636209366079954659965017957689652962086691638433510525888
-207:
-0.6577920842830206535118453273055800247916112037643622953206746235657530224497080235242195075104228218
-208:
-0.7166561814767082834337327253304109020954158590890644634178329951454865668750291896911713680918125652
-211:
-0.5181674884301134453796110480548637844945731362566269214866325926466539693204544753355969182414664800
-212:
-0.7025228512813055727256858188637116143891966572002636814049515503845687217642226712537267451720259035
-215:
-0.9701866450990100028554716846864929254359218627748495118170878114884873129503272579903890434138819961
-216:
-0.7839581911893335015425201690928904190782391576757536219103680964201753397595061986033911848913115694
-219:
-0.6199579638857112658422643944313760589379545721612950731393177368312610957948712022980119815725506704
-220:
-0.7089085239817010624768294256646914711635182577064173800709966532791561564847097826942259748104400751
-223:
-0.7115887984151674877726777706228242749377768941236913105503943941240044568527788803680635944401214551
-224:
-0.9020579334820643716519580063640226743510884494592312541763469244710645000429120405886463813954974844
-227:
-0.6990175133274220078196261563432183786778233214752334602578402440643473560078414377821255158035948404
-228:
-0.4243420451651863740961066216700654260456642537912977534348561310649539264372313423744977182250095710
-231:
-0.9105780415261989908305555317691535592849213006698143935620788372308859758027095430385461561532623051
-232:
0.3110060812332979889140745147768076435304631933565210839226072422230821575497988372735873953247584812
-235:
-0.04592079832905624118506817223576339474316952274271349615071611778467284479248384572382903031334122829
-236:
-0.9115678664499480817305897969399166665264653109513726842799480746773185038305668083674115872214389634
-239:
-0.9545767130102608230155631539270515523612832332186946270759516783380430985346861940549247402360442369
-240:
-0.5414365425804279375555852772604789999120407927203225406683828664434385548571913203867008398102340475
-243:
-0.4666412039083970439046784560261923601551548753067549289722128518789024057856329212727071578634992403
-244:
-0.7261226985986953330233499530524493428061376348132766102271769326593705606326842013170728984186894863
-247:
-0.5484335780904054707646639265524650220221090143895243050562357854215884568583148204710558618175705050
-248:
-0.7859786934704732796782020111248922884278923706571924255843219395200313481717120287857170807221167499
-251:
-0.8135558218127785707563022954811401345110067401106903241827371814418233750889794182973501428196797211
-252:
-0.6643892638724506658432225256275092374747974452145352898311958221002983031985542778520315564373206027
-255:
-0.8762796560630123282716005857771132568026119396848773118061512007073140552429629277821237474328584832
-256:
-0.5740640465665676269964659542554160643713886966500388536192599897517292950964892816032428857673693355
-259:
-0.6101284021419282381896288303225374604191883629104648590962145963988905621131104002802825322815923462
-260:
-0.7563671704124534370520289254198129268994278743840223572235035635678344192802362660193591601580362392
-263:
-0.8708907875723479173466080177319791023008324972418851611325847544093763475539575610009952328775559895
-264:
-0.7889026349978192597642213020870126481201162132827223287390026221040833817161998697607869303949006587
-267:
0.2384288909875939765555674646174272123398364658409371316476864015322934078052129222286803965985636795
-268:
-0.2840317563010989711916241027030892416890307513671913162524279390679564852089053680021853349979458022
-271:
-0.8449139673811912250504312062628334651842398694573271116191420849527528490980608484916393234796383689
-272:
-0.8423084564307857143285894202143314026247079342053449481697344027536522408685005099829728764096152438
-275:
-0.4178276863120168983862775880168125009142113479908622249913066916475400197621648091369343467198547225
-276:
-0.7677373630779326191965919189002921750120248992146165271237780271928812104900332528190201372788735950
-279:
-0.8723400113526955493640232690550481890846141146472382175196975555414564209363642773365622168587680254
-280:
-0.3503810702542715656691326894787768877976060931461691782571397715629550564751685536036153703540095459
-283:
-0.3669742501222557363215354400743448786937781735169328515933976085784871317105702966182809122543712268
-284:
-0.7589553795121086416902347247705376160000184916881313406391803118184098892110416481561435578343332961
-287:
-0.8660599976614781522008909007288267379934365587120890447356564059538766534913363535655081722169132761
-288:
-0.5478136865857845834758871086115138830012336609833995338087577350657940785779826957645226710293394261
-291:
-0.4347044929827976326775060502659521082974967841308487611013809867094485633938832354254671987172287459
-292:
-0.3403190091492902947330262033156329508424767861375825344068822905899358026086581101289369098527353929
-295:
-0.6347250201752012738962518422285778554366090789067921763084236831192639364338916611230365821806722687
-296:
-0.8041772764636743635907969107682656530612211703871274392393015690594239188135738553135820713543116481
-299:
-0.8033851384266806097821709294176287671344470109089359969134154899761516936336770210512005217717272164
-300:
-0.6691135941900135718867440645363981559154192797971227195526458126344393676137159658078308977437167690
-303:
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-0.6590489520509465381425841095744785615853441311698372979862150159903986249831325317852905380397619354
-923:
-0.3706598713031966818171629004183060413629251375417170981926953829561917821705782805961788174064228147
-924:
-0.5640044512462263361219394710400652752471712334896867665017388324841891648178621852356144926550529614
-927:
-0.6311529089515943276347920870902104056045888860634693745563692463259575668819755801543357681535103659
-928:
0.4842928763732843162683825451413517855493382269465848974527772445964305630422225161750532270738631531
-931:
-0.1133033301547282858538559243428308565321218827727249639890193597510186512920372226547235706198096720
-932:
-0.2973844923929128075030383616455451659835946431733445188357895546734835154427531241701515337090773960
-935:
-0.6715636891153132641162631769918563941124989115499009600929951890795674951450236830097118007966435441
-936:
-0.4527888548858320528869089701358906746602311160980568762385725367870621461142315754762568313594906064
-939:
-0.2476733221916179683166580312984740450531469468065405482250164859684513760373329168798419445530122225
-940:
0.06960373176426797705113718134059936660274716631732904620273055046422609220246527354381485751939521963
-943:
-0.4640242567950861111631984873329337175166230071880129073063846273435032781670716602244269967011809315
-944:
-0.6805188062632996452581790897871911438346319328312875995730547381795206298406685698321238115559660676
-947:
0.4980831971336099210849723544176482223450810399242949538899227852609535752093779972903250605826584845
-948:
-0.4030438935448529574809802760883346452430497366104274699237955411499961704420595068645053246359741732
-951:
-0.6589137724760737036743908057350211292636787380305907485580727106644480572688507725049176211247344192
-952:
-0.1855376051462435021776596364559296168264366507549660913006625392584021245680978604827381657383751178
-955:
0.4859332884060527499843788453866100258089126126895239391347107928792707986012520094532223683272679061
-956:
-0.6080031227302881683069470931979632683235331660385670000156116735913462875498388362519930767378348931
-959:
-0.7769437876320077956205175541344732710349701676644617754305562026677735917644511076893953771218233696
-960:
-0.1948629523004552828469692165313907158742907255401949136080428616967417438723439625837691763120247037
-963:
-0.2395017634095076964886174182035915449937004852282178615782374717429688080522038767380660155767086608
-964:
-0.4365141249632869207822020167766208860915900558292354552748784746841197845572350146104929540622525418
-967:
-0.09978778568062457888566233625456084099720827909454687534881042238140682432590089581053381709192592805
-968:
-0.07145898966064048449684596845475622743300054651612948559699104969260281872309250156526902618399274664
-971:
-0.5935842029604815637051159064045725283709879606219743088059800340679516588340572173483862746785822685
-972:
-0.3511166738150728256684731024498295988092381862467123866187661836300034687906838020050632700307627924
-975:
-0.3388694223792175471665358038259412052669894563931524235198803007139671169468541559608195449848124319
-976:
-0.5528359034587090056690419226879052007872626012232127966970069302860221551402605224156070666695848145
-979:
-0.3212041863486899144982554879144388927112107728536408305165370605936459717412137611907796192525909550
-980:
-0.2167803885443314495414539911481857551896222251552302700929003557211657360699396582552942632836614358
-983:
-0.6291433052174012169702745120221523680652905745285336827593562103845574967164241294526834056267636961
-984:
-0.3233068298844457482039778229415029420090779155410813186502318556747191824587947081709922935607237298
-987:
-0.3043062447110531318548763280019027946714889958019766364340743687050253740508243415766078353107035666
-988:
-0.2018599878104328160560478658233767379843589472093966779958957806748916458734674626681241983193611612
-991:
-0.4360944005882423972739578144745550358255391189247466426913836764699812518702759350614909728696737471
-992:
-0.6126918983304869523238939807603481464090173370671286120541519371466829426792883498842512489730120780
-995:
-0.07166570583605090751456926235802648945403066203217088573646193391075273112792906530526728291507049163
-996:
-0.3096200591998741455226439347151224153894048713329471026072593579386915070617605055141565444035345950
-999:
-0.6580461140882325679762502864623377927611691616490985391785694575219909289431509667246556441021499761
-1000:
-0.3545893715495336469237819505545225649564209392626979066269770841147023381263602979885928029309873056
Definition
The stable Faltings height $h_{\mathrm{st}}(\Delta)$ of an elliptic curve with endomorphism order $\mathcal O_\Delta$ is its Faltings height after extension to everywhere semistable reduction, with metric $(i/2)\int\omega\wedge\overline\omega$ [1], §1.
Parameters
$\Delta$
—   order discriminant ($\Delta<0$ and $\Delta\equiv0,1\pmod4$)
Formulas
(1)
$h_{\mathrm{st}}(\Delta)=\tfrac14\log d-\log b_D+\tfrac12\log f-\tfrac12\sum_{p\mid f}c_p(D,f)\log p$, where the sum is over prime divisors of $f$ and $c_p(D,f)=\dfrac{1-\chi_D(p)}{p-\chi_D(p)}\dfrac{1-p^{-v_p(f)}}{1-p^{-1}}$. Here $v_p(f)$ is the exponent of $p$ in $f$, and $b_D$ is the field period constant. This is [1], Theorem 1.3; $c_p$ is the negative of that paper’s $e(p)$.
(2)
$\log b_D=\tfrac12\log\pi+\dfrac{w_D}{4h_D}\sum_{a=1}^{d-1}\chi_D(a)\log\Gamma(a/d)$ [2], equation (2.1), where $\Gamma$ is Euler’s Gamma function.
(3)
$h_{\mathrm{st}}(\Delta)=-\log(2\pi)-\dfrac1{h(\Delta)}\sum_{[a,b,c]}\log F(\tau_{a,b,c})$, with the forms and $F$ defined in (10). This follows by combining [1], Theorems 1.3 and 5.1; choosing $-b$ in place of $b$ permutes the complete class set.
(4)
$h_{\mathrm{st}}(D)=\tfrac14\log|D|-\log b_D$ when $f=1$, since the conductor sum is empty.
(5)
Deligne’s geometric height satisfies $h_{\mathrm{geom}}=h_{\mathrm{st}}+\tfrac12\log\pi$ [1], Remark 1.4.
Comments
(6)
$\mathcal O_\Delta$ is the imaginary quadratic order of discriminant $\Delta$, and the endomorphisms are taken over the algebraic closure. Write $\Delta=f^2D$ uniquely with $D<0$ a fundamental field discriminant and $f\geq1$ the conductor. Set $K_D=\mathbb Q(\sqrt D)$ and $\mathcal O_\Delta=\mathbb Z+f\mathcal O_{K_D}$. The stable height depends only on this order, so conjugate singular moduli and twists do not require separate rows. Hilbert class polynomials describe the singular moduli for the same order index.
(7)
Let $d=|D|$, $\chi_D(a)=(D/a)$ be the primitive Kronecker character, and $h_D$ be the ideal class number of the maximal order. Its unit count is $w_{-3}=6$, $w_{-4}=4$, and $w_D=2$ otherwise. These are maximal-order invariants even when $f>1$. All logarithms are natural.
(8)
This is the Silverman/Faltings normalization. This metric agrees with the stable rows for elliptic curves over $\mathbb Q$. The unstable height of a chosen model can differ.
(9)
The positive constant $b_D$ is the Chowla–Selberg period constant in the normalization of [2], equation (2.1). It is indexed by fields, while this table is indexed by orders. The conductor correction in (1) accounts for this difference.
(10)
For a primitive reduced positive definite binary quadratic form $[a,b,c]$ with $b^2-4ac=\Delta$, set $\tau_{a,b,c}=(-b+i\sqrt{|\Delta|})/(2a)$, taking the positive square root. The representatives satisfy $|b|\leq a\leq c$, $\gcd(a,b,c)=1$, and $b\geq0$ when $|b|=a$ or $a=c$. Their number is the order class number $h(\Delta)$. Write $F(\tau)=\sqrt{\operatorname{Im}\tau}\,|\eta(\tau)|^2$, with $\eta(\tau)=e^{\pi i\tau/12}\prod_{n\geq1}(1-e^{2\pi in\tau})$ [3]. Rectangular CM eta values give individual values used in special cases of the class product.
Programs
(P1)
PARI/GP
\p 120
cmheight(Delta) = {
  my(D=quaddisc(Delta), f=sqrtint(Delta/D), d=abs(D));
  my(h=qfbclassno(D), w=if(D==-3,6,if(D==-4,4,2)));
  my(S=sum(a=1,d-1,kronecker(D,a)*lngamma(a/d)));
  my(H=log(d)/4-log(Pi)/2-w*S/(4*h)+log(f)/2);
  forprime(p=2,f,
    if(f%p==0,
      my(v=valuation(f,p),chi=kronecker(D,p));
      H-=(1-chi)/(p-chi)*(1-p^(-v))/(1-p^(-1))*log(p)/2
    )
  );
  H
};
cmheight(-23)
References
[1]
Adrian Barquero-Sanchez, Lindsay Cadwallader, Olivia Cannon, Tyler Genao and Riad Masri, “Faltings heights of CM elliptic curves and special Gamma values”, Research in Number Theory 3, article 13 (2017), §1, Theorems 1.3 and 5.1. (doi)
[2]
Wen-Ching Winnie Li, Ling Long and Fang-Ting Tu, “Computing Special L-Values of Certain Modular Forms with Complex Multiplication”, SIGMA 14, 090 (2018), §2.1, equation (2.1). (doi)
Links
Similar tables
Chowla_Selberg_period_constants —   holds the field-indexed $b_D$ entering the stable height through (1)
Faltings_heights_of_elliptic_curves_over_Q —   holds stable and unstable heights indexed by rational elliptic curves; its CM stable rows agree for the same endomorphism order
Hilbert_class_polynomials —   uses the same order discriminant to encode conjugate singular moduli algebraically
Dedekind_eta_values_at_rectangular_CM_points —   holds individual rectangular eta values, while the stable height uses the product over every ideal class
Values_of_the_Gamma_function_at_rational_numbers —   holds the individual Gamma values in the field-period formula
Data properties
Entries are of type: real number
digits: 100 (Unknown key)
How well the digits are known: proven
Table is complete: no (it holds every imaginary quadratic order discriminant with $|\Delta|\leq1000$, in increasing absolute value: 305 maximal and 195 nonmaximal orders, with conductor at most 18. This arithmetic selection includes class number greater than one and split, inert and ramified conductor primes, including prime-power and composite conductors. The noncanonical bound keeps each Gamma sum below 1000 terms and permits a separate eta-class-product check for every order. The attached argument-selection.md explains the coverage and bound)
How they were obtained:

The generator uses Arb real balls at 429 bits, with 96 extra working bits beyond 100 significant digits, to evaluate the logarithmic Gamma sum and exact rational conductor correction. The maximal-order class number is obtained from the exact finite character sum. The smallest measured relative accuracy is 408 bits; no precision escalation was needed. Every computed ball lies inside the interval denoted by its written decimal, whose uncertainty is one unit in the last place.

more

All 500 orders were independently computed from the eta class product using primitive reduced forms and 512-bit real balls. The product is truncated after 80 terms. For $r=|e^{2\pi i\tau}|$, the omitted contribution to $\log F(\tau)$ is bounded in absolute value by $2r^{81}/((1-r)(1-r^{81}))$, obtained from the power series of $\log(1-z)$; the largest bound is less than $8\cdot10^{-192}$. The eta enclosures retain at least 493 relative accuracy bits and also fit entirely inside the written decimal intervals. Every comparison requires finite, narrow enclosures. The square-lattice control uses an independent arithmetic-geometric mean computation of $K(1/2)$, and perturbing any height by $10^{-90}$ is rejected.

An independent exact decomposition recovers the same field and conductor for every order. Reduced-form counts agree with PARI class numbers and the ring class number formula throughout. The conductor formula is checked on all 500 orders, the period relation against all 305 maximal-order entries of T669, and the field-height relation against all 305 imaginary-field Euler–Kronecker constants in T230. Further checks use all 150 degrees in T129, twenty CM stable-height rows in T211, the rectangular eta values at $r=1,4$ in T506, and the square-lattice elliptic integral in T296. The attached checks repeat enclosure and conductor/period-identity tests on an authenticated export of the stored draft.



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