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Elliptic trilogarithm L32 at CM torsion points

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Numbers
$\Delta$
$N$
$r$
$s$ 
$\mathcal L_{3,2}^{E_\Delta}(u)$
-3
1
0
0:
0.2516553831641618343396078379090784628700764659822877556906726797997743048027634743916416421569720580
-3
2
0
1:
-0.2527674973076969394214545773359882978730488508964085788820457452227375561243292873039040046315541030
-3
2
1
0:
-2.485382952711650560220099171637627492796991630856768319944956825095207997439233391021968984531636134
-3
2
1
1:
-2.485382952711650560220099171637627492796991630856768319944956825095207997439233391021968984531636134
-3
3
0
1:
-0.1258252821574562804000601561059257068234767114624412669212800648793843259223149664337889804811026541
-3
3
1
0:
-2.039574468801066734374311352660862456996677645970764586390482328227303284005222072464346353259409716
-3
3
1
1:
-2.039574468801066734374311352660862456996677645970764586390482328227303284005222072464346353259409716
-3
3
1
2:
-1.874033346515298988017894894300158398312949848882355261498297074800217327343400611838675477901958625
-3
4
0
1:
0.0005560466307136269171519775135090878837175881577008492169673833339017984323150120116315663989789041457
-3
4
1
0:
-1.512012120108374043411061502120353597646596383759244911741178791955821486199305380286254314050546649
-3
4
1
1:
-1.512012120108374043411061502120353597646596383759244911741178791955821486199305380286254314050546649
-3
4
1
2:
-1.316056681659520618669924415424067057110264951659465829253918253962740924680960525368547180054456531
-3
4
1
3:
-1.316056681659520618669924415424067057110264951659465829253918253962740924680960525368547180054456531
-3
4
2
1:
-2.549547916473114086398307320770224843275086076081366622579200544268405404148217904701674938016471831
-3
5
0
1:
0.07838567912613599776957538799218419414805264940481476596859969068638582354477123801145297753395855818
-3
5
0
2:
-0.2042133706629714377843503016034734803964307917914785110959610522535346943427892383870703683156518628
-3
5
1
0:
-1.123491179485740022875074292432038656582681724889430041086789066088849817936160109927985044806345661
-3
5
1
1:
-1.123491179485740022875074292432038656582681724889430041086789066088849817936160109927985044806345661
-3
5
1
2:
-0.9573522930105722493092426894836428152034439426993821857388706400926511331844973930222325942461887207
-3
5
1
3:
-0.9888187224600263245941785692717436435691267481780081530236105487546546863473322236812497058423135016
-3
5
1
4:
-0.9573522930105722493092426894836428152034439426993821857388706400926511331844973930222325942461887207
-3
5
2
0:
-2.322802715265346460781826666695242915567059433576436856806288262875424236583141092427056636641363720
-3
5
2
1:
-2.475020387136071965290282846945767211588462876796235737522416834312951038287946158768689389422105065
-3
5
2
2:
-2.322802715265346460781826666695242915567059433576436856806288262875424236583141092427056636641363720
-3
5
2
3:
-2.239811367586780044679703945759433275059935372827099884831700031355012524207020290417981750217009209
-3
5
2
4:
-2.239811367586780044679703945759433275059935372827099884831700031355012524207020290417981750217009209
-3
6
0
1:
0.1263813392290277652622945959049130732985339516893658803832174067963945975271448967397267456492452358
-3
6
1
0:
-0.8463523865622247177904172201569758830926223707560386508209872666797854042785518502286140757325781443
-3
6
1
1:
-0.8463523865622247177904172201569758830926223707560386508209872666797854042785518502286140757325781443
-3
6
1
2:
-0.7136188506229222319648098250844076136546001743848521539460971198493931775610789785099091601279690016
-3
6
1
3:
-0.7691727249938634925168826679632157051161617846090833633861826189127256951414642198778924409969721369
-3
6
1
4:
-0.7691727249938634925168826679632157051161617846090833633861826189127256951414642198778924409969721369
-3
6
1
5:
-0.7136188506229222319648098250844076136546001743848521539460971198493931775610789785099091601279690016
-3
6
2
1:
-2.202029154225557979163906221285500410270562027691758647988037839764373471771015047655285813394690356
-3
6
2
3:
-1.889585902200988423018913639563596005358182373280033948003923253800334259976790839099344460402989759
-3
6
2
5:
-1.889585902200988423018913639563596005358182373280033948003923253800334259976790839099344460402989759
-3
6
3
1:
-2.533402121484406119534598260502451614747700590607882166098106121850089587743066067734247876182757786
-3
6
3
2:
-2.533402121484406119534598260502451614747700590607882166098106121850089587743066067734247876182757786
-4
1
0
0:
-0.1486912576145472438467008934590611467384496376905544624348386721505187015698531075018758586861741922
comment: The origin on the square torus. (4) gives its Dirichlet-beta expression.
-4
2
0
1:
0.1484159037777092937749280898628063515521461865262757465017998020445407772703070703996243209224924789
-4
2
1
0:
-4.762184591419045509134700126563250345201327061986403285014770849577870339497699699159160511141381860
-4
2
1
1:
-3.026282039222708395624281059980608946853304037743233538197034474305380140961644050005732173917201523
-4
3
0
1:
0.07434537170754016754830291735775965662581614534226834171448011906029541247340997383683079976760941858
-4
3
1
0:
-3.926262431277793865237470706904725663211861315846539336057753918724093292885566593436189870055169547
-4
3
1
1:
-2.634521439678280456297084805509895290354031356873683169567960151365229484254799671876451908496756959
-4
3
1
2:
-2.634521439678280456297084805509895290354031356873683169567960151365229484254799671876451908496756959
-4
4
0
1:
0.0001376764382999605164752132790357937047614979734046709385054615861362871468756973839805327898082013330
-4
4
1
0:
-2.981660483902517812210622304719073520893844806085050009470732830319379643315342364683334370942753580
-4
4
1
1:
-2.127034919970158035693787257942625259317412037726771327021348286131985941151709380473277075407505482
-4
4
1
2:
-1.489397175062340398365754594489007172534482147398888862924180788853653513805083352407398322025279926
-4
4
1
3:
-2.127034919970158035693787257942625259317412037726771327021348286131985941151709380473277075407505482
-4
4
2
1:
-3.857370273503563820729148585433859400038904295393258664393847749306368054671300568279865121749857416
-4
5
0
1:
-0.04579400668143639122966650142181738535966586209022435293018232060101844513736843721814020361225332237
-4
5
0
2:
0.1201396354878134183890468476917976774739590685680602690075533430458334854651318183259133485535221103
-4
5
1
0:
-2.314732484018476979114554546878780910365477233291731756225890678403057876199828516239364796772654094
-4
5
1
1:
-1.726908682475415647999980690728456040307987284961256482151589676558461028148020687597036686108752578
-4
5
1
2:
-1.086004342891634972793070368333263744010188554982397367749233908868056554106904727129686431744337696
-4
5
1
3:
-1.086004342891634972793070368333263744010188554982397367749233908868056554106904727129686431744337696
-4
5
1
4:
-1.726908682475415647999980690728456040307987284961256482151589676558461028148020687597036686108752578
-4
5
2
0:
-4.453123317780203509233896276910642055730495181777326494684998551187359355300601445286365820473494453
-4
5
2
1:
-3.815523422910517357744459685019335029650477760482998078339956718880759152059666995880235389997654418
-4
5
2
2:
-2.887733662403514145342892870643229577948998925640208634122227161987257967388043265011360179474934279
-4
5
2
3:
-2.887733662403514145342892870643229577948998925640208634122227161987257967388043265011360179474934279
-4
5
2
4:
-3.815523422910517357744459685019335029650477760482998078339956718880759152059666995880235389997654418
-4
6
0
1:
-0.07420769478912286685179177879270357670654895656603646478707532490323290711624015744244119638355997682
-4
6
1
0:
-1.853249622753149087171264824958472597189801782872363267986686525168058458887169010713846380165628626
-4
6
1
1:
-1.429085493244264985672838648586985731515008792340895348432312108931359477594810086780057962335421693
-4
6
1
2:
-0.8919438740682143880574814683990384522262415844331892067838810528466448403505742051794380571902456256
-4
6
1
3:
-0.6841202984478521201159980552572552442288971273492326186856725192398756315404915956732119858917673053
-4
6
1
4:
-0.8919438740682143880574814683990384522262415844331892067838810528466448403505742051794380571902456256
-4
6
1
5:
-1.429085493244264985672838648586985731515008792340895348432312108931359477594810086780057962335421693
-4
6
2
1:
-3.454648723003765327915945045656903149049548247041929991143971250649672298229942463524840307252486511
-4
6
2
3:
-2.269502425415608437653712856381076711728727244364776534820460042500594669274590957927474489551948457
-4
6
2
5:
-3.454648723003765327915945045656903149049548247041929991143971250649672298229942463524840307252486511
-4
6
3
1:
-4.299739417720138673587181974125953079984146772197201095367921453222834955795292664117221058838006763
-4
6
3
2:
-3.433381086346348629091332201875011886334988521714740619666997837120752425809951168209636102754861872
-7
1
0
0:
0.03389590883944323124159052670872163447490623244434439016926533109550396636942632906823444814621968694
-7
2
0
1:
-0.03390424232803090317529607403860418078623580436691759543590090628251491695489787115608043615992928541
-7
2
1
0:
-8.968325723509257568824421531021086611378085593810256258853498393231385106658601105859730192333597250
-7
2
1
1:
-8.968325723509257568824421531021086611378085593810256258853498393231385106658601105859730192333597250
-7
3
0
1:
-0.01694795339643756636424809927575278715621941897419222199846405425288569179626217787279024983381958386
-7
3
1
0:
-7.260448722012240746922324970588488824133716910947253539261429825057102051009323788405011867612880815
-7
3
1
1:
-7.260448722012240746922324970588488824133716910947253539261429825057102051009323788405011867612880815
-7
3
1
2:
-6.745526126764864440956782183031722057156278201881879623128363043584876005158204567657870936744616004
-7
4
0
1:
0.000004166744042534121587557139097985748818387802524863693557312969104143306016335498088060162094667468883
-7
4
1
0:
-5.370827078128150955518861037966529603799945965148979098524651215369530421613606341899760080356977924
-7
4
1
1:
-5.370827078128150955518861037966529603799945965148979098524651215369530421613606341899760080356977924
-7
4
1
2:
-4.722780516168988049280664773935724311465428392904388844090010430996221294466233964369888686043451243
-7
4
1
3:
-4.722780516168988049280664773935724311465428392904388844090010430996221294466233964369888686043451243
-7
4
2
1:
-8.976809075893528205714972686918541597190211522175475473381019273466247085672893737429780607040815338
-7
5
0
1:
0.01047906967006353485009656874266254848953369841626581491816902016051063775065077509587819726547735386
-7
5
0
2:
-0.02742702408978508875526411819471643409739410530186334643474570019757552878020231858668093728781090338
-7
5
1
0:
-4.061804482585636298833157435990440316506174130649606095643174418366249645627697335858583391433120734
-7
5
1
1:
-4.061804482585636298833157435990440316506174130649606095643174418366249645627697335858583391433120734
-7
5
1
2:
-3.497258916253874800940332279673255284947122459127693423907307419075028960497853063801421253454544777
-7
5
1
3:
-3.257011585735815818956481277600115006509516512194573178581450755721065315715023210384902198353148343
-7
5
1
4:
-3.497258916253874800940332279673255284947122459127693423907307419075028960497853063801421253454544777
-7
5
2
0:
-8.332897908837560555650176106805147874602103868741887398333023708572251619387160622088632589356437132
-7
5
2
1:
-8.504111383149456052192695307401851255196801656797242726712001919650726670179001850272555105013434386
-7
5
2
2:
-8.332897908837560555650176106805147874602103868741887398333023708572251619387160622088632589356437132
-7
5
2
3:
-8.087840821632925466889006381820168642231370005905290926690642083265927794697893474381352346186383753
-7
5
2
4:
-8.087840821632925466889006381820168642231370005905290926690642083265927794697893474381352346186383753
-7
6
0
1:
0.01695212014073140231594452301377118240332794881582033975033603904118334261965551010097712111980171891
-7
6
1
0:
-3.168181180823364867914440401879941548996677281399438397347329107975401211128276646569500930209150189
-7
6
1
1:
-3.168181180823364867914440401879941548996677281399438397347329107975401211128276646569500930209150189
-7
6
1
2:
-2.711004456973972257493221194401363961986850202414624988941331662082694135138438604170730010256224324
-7
6
1
3:
-2.428511037828284819927494256742358043022767796148130140911634865138776083393531160802650543005733819
-7
6
1
4:
-2.428511037828284819927494256742358043022767796148130140911634865138776083393531160802650543005733819
-7
6
1
5:
-2.711004456973972257493221194401363961986850202414624988941331662082694135138438604170730010256224324
-7
6
2
1:
-7.463980248695370771688917214972716710845318261413512125287013087569611881349499701989501761646548370
-7
6
2
3:
-6.903104997556133369359267854485569967566494556896163127810912352516653322716229687398454610792914401
-7
6
2
5:
-6.903104997556133369359267854485569967566494556896163127810912352516653322716229687398454610792914401
-7
6
3
1:
-8.974687456405183422467852344803096384861195351119673996883718346387588114633744606385374709230948026
-7
6
3
2:
-8.974687456405183422467852344803096384861195351119673996883718346387588114633744606385374709230948026
-8
1
0
0:
-0.02186004257366863131864610101852182915506427909132164268588803170080264450926138447311229194641476386
-8
2
0
1:
0.02185702024569872718925072630430344007933325839232152978858466984992327820110055315671647232245707048
-8
2
1
0:
-11.43004941466145930632600574208274970718753726542631190229886068918120229193431557527242275601015201
-8
2
1
1:
-10.50016107364030406213173492583911872817430457910346432253844940084528481554798492427841549949153915
-8
3
0
1:
0.01093002107777358222029338390284767756239320486109214665122724182197486504507135001689239037476756067
-8
3
1
0:
-9.222595970108205135695089035788797143324191442450310868455855432317667116884315143658098040949781288
-8
3
1
1:
-8.381752793793637780603628680427105670802453586862694527130831642271777471693517638332901703695686984
-8
3
1
2:
-8.381752793793637780603628680427105670802453586862694527130831642271777471693517638332901703695686984
-8
4
0
1:
0.000001511163956029727701502782722985384301965322631290063518364655536418822703114071731756345086590856642
-8
4
1
0:
-6.791419779746630296055472153113076769489413907626934438259794462360052248282369168695192840454388483
-8
4
1
1:
-6.137447192099098032572602062210309012298095735057953001924438448540871754137535553954639969252982485
-8
4
1
2:
-5.599710647079210174614015823337995127483431648279327513059133845763680944540993270588338005116403095
-8
4
1
3:
-6.137447192099098032572602062210309012298095735057953001924438448540871754137535553954639969252982485
-8
4
2
1:
-10.95964363362640174620778242068541323894878870575156406237151600290037891537913853730162019837430203
-8
5
0
1:
-0.006753434964517218231989885503264798342559018041779673479910217407601842354092454010030649019347508474
-8
5
0
2:
0.01768345625135152989007576645678080391435060446488099515825615486766152542774707712440769871832607215
-8
5
1
0:
-5.119361603307471509619402178370788845305097248507346208192742805354568006013629743813165732135745782
-8
5
1
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Definition
The real elliptic trilogarithm $\mathcal L_{3,2}^{E_\Delta}(u)$ is the logarithmic elliptic average with its quartic correction in (1), evaluated at the torsion point $u=(r\tau_\Delta+s)/N$ of the principal marked CM torus [1], section 1.
Parameters
$\Delta$
—   CM order discriminant ($\Delta<0$ is an imaginary quadratic order discriminant)
$N$
—   exact order of the torsion point ($N\geq1$)
$r$
—   $\tau_\Delta$ coordinate numerator ($0\leq r<N$; together with $s$, it satisfies (7))
$s$
—   real coordinate numerator ($0\leq s<N$; together with $r$, it satisfies (7))
Formulas
(1)
$\mathcal L_{3,2}^{E}(u)=\sum_{n=0}^{\infty}J_3(q^n x)+\sum_{n=1}^{\infty}J_3(q^n/x)+\frac{\log^2|x|\log^2|q/x|}{4\log|q|}$, where $x=e^{2\pi i u}$ and $J_3(z)=\log^2|z|\log|1-z|$, with $J_3(1)=0$ [1], section 1.
(2)
$\mathcal L_{3,2}^{E}(-u)=\mathcal L_{3,2}^{E}(u)$. For real $q$, conjugating $x$ also preserves the value: $\mathcal L_{3,2}^{E_\Delta}((r\tau_\Delta+(br-s))/N)=\mathcal L_{3,2}^{E_\Delta}((r\tau_\Delta+s)/N)$. Coordinates are interpreted modulo $N$.
(3)
For every positive integer $M$, $\sum_{r,s=0}^{M-1}\mathcal L_{3,2}^{E}((r\tau+s)/M)=M^{-2}\mathcal L_{3,2}^{E}(0)+(M^2-M^{-2})c$, where $c=\log^3|q|/120$. This follows from finite-character orthogonality in the Eisenstein-Kronecker expansion [1], equation (2.10). The sum includes all $M$-torsion points, including those of smaller exact order.
(4)
For $\tau=i$, $\mathcal L_{3,2}^{E}(0)=\frac{1}{\pi}Z_{\mathbb Z^2}(2)-\frac{\pi^3}{15}=\frac{2\pi}{3}\beta(2)-\frac{\pi^3}{15}$, where $Z_{\mathbb Z^2}(2)$ is the square-lattice Epstein zeta value and $\beta(2)=L(2,\chi_{-4})$ is the Dirichlet beta value. This is the square-lattice specialization of [1], equation (2.10).
Comments
(5)
Put $b=0$ when $4\mid\Delta$ and $b=1$ otherwise, $\tau_\Delta=(-b+i\sqrt{|\Delta|})/2$, $E_\Delta=\mathbb C/(\mathbb Z+\tau_\Delta\mathbb Z)$, and $q=e^{2\pi i\tau_\Delta}$. The square root is positive. The marking is part of the convention and is not replaced by a modularly equivalent representative. In particular, $q$ is negative for odd $\Delta$.
(6)
All logarithms are natural. Write $x=e^{2\pi i u}$ and $J_3(z)=\log^2|z|\log|1-z|$ for $z\ne1$, with the continuous value $J_3(1)=0$. The logarithms act on absolute values, so no logarithm branch is selected. On $|z|=1$, the value of $J_3(z)$ is zero. The defining annulus is $|q|<|x|\leq1$.
(7)
The condition $\gcd(N,r,s)=1$ makes $N$ the exact order of the point. The origin is $(N,r,s)=(1,0,0)$. Among $(r,s)$ and $((-r)\bmod N,(-s)\bmod N)$, the lexicographically smaller pair is listed, with self-inverse points included once. Omitted opposite points have the same value by evenness. Signs are retained, and the origin and all points of order two are included.
(8)
The current discriminants are exactly the thirteen CM orders with rational singular modulus, including nonmaximal orders [2]. Thus $\Delta$ is an order discriminant and need not be a fundamental field discriminant. For these orders, the principal lattice represents the unique ideal class.
(9)
The elliptic trilogarithm $\mathcal L_{3,1}^{E}$ is the other real weight-three component. The two components are different linear combinations of the Eisenstein-Kronecker series $K_{1,3}$ and $K_{2,2}$, with the constant $\log^3|q|/120$ present in $\mathcal L_{3,2}^{E}$ [1], Corollary 2.8. One component does not determine the other. The Bloch elliptic dilogarithm and elliptic dilogarithm companion $J$ use the same marked CM lattices and torsion coordinates at weight two.
(10)
Both real elliptic trilogarithms occur in the K3 Mahler-measure relations of [1], Theorem 1.2. The torsion groups represented here include the orders in those relations, but a separate Mahler-measure identity is not asserted for every listed point.
(11)
The quartic logarithmic correction is part of the definition, not a rounding adjustment. Consequently the torsion distribution contains the constant term $c=\log^3|q|/120$ in (3). The origin and all points of order two are retained; evenness does not force them to vanish.
Programs
(P1)
Sage
# Download the attached generate.py into this directory first.
from generate import Table
g = Table()
print(g.value({'Delta': -4, 'N': 1, 'r': 0, 's': 0}, 100))
References
[1]
Detchat Samart, The elliptic trilogarithm and Mahler measures of K3 surfaces (2014), sections 1 and 2. (arXiv)
Links
Similar tables
Elliptic trilogarithm L31 at CM torsion points —   gives the other weight-three component on the same marked tori; both are expressed through Eisenstein-Kronecker series
Bloch elliptic dilogarithm at CM torsion points —   gives the odd weight-two elliptic polylogarithm on the same marked CM lattices
Elliptic dilogarithm companion J at CM torsion points —   gives the logarithmic weight-two component, with a cubic Bernoulli correction and different parity
Rational singular moduli —   identifies the thirteen class-number-one CM orders in the current coverage
Epstein zeta values —   provides the square-lattice value in (4)
Dirichlet L-values —   contains the Dirichlet beta value in (4)
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds 494 values on the principal marked lattices of all thirteen class-number-one orders, $\Delta=-3,-4,-7,-8,-11,-12,-16,-19,-27,-28,-43,-67,-163$, at exact orders $1\leq N\leq6$, with every torsion direction represented modulo inversion. The origin and all 2-torsion points are retained. This arithmetic reference covers rectangular and rhombic CM tori, the orders $2,3,4$ in trilogarithm identities, the next prime order $5$, and the first mixed-prime order $6$. The cutoff six and lexicographic representatives are noncanonical choices independent of numeral notation; no separate application is claimed for every combination. Higher class numbers and higher torsion orders remain outside this coverage)
How they were obtained:

Every value is enclosed by Arb evaluation of the two logarithmic series and the quartic correction. The marked CM arguments are enclosed from exact integer coordinates. For $|z|<1$, the bound $|\log|1-z||\leq |z|/(1-|z|)$ gives explicit bounds for both omitted tails through geometric sums weighted by a quadratic. Those bounds are added to the computed balls and justify the 100 significant digits recorded. Unit-circle logarithmic factors are treated exactly, including $J_3(1)=0$. With 192 extra working bits, the minimum measured relative accuracy was 430 bits; between 7 and 58 terms on each side were used.

more

Every entry was recomputed with 30 extra decimal working digits. A separate mpmath calculation expands the logarithm and sums the geometric index analytically before summing the logarithm index. Its values at 150 and 190 working decimal digits differed relatively by less than $3.5\times10^{-134}$; the higher-precision values differed from the higher-precision Arb midpoints by less than $9.1\times10^{-161}$. The nonzero geometric-moment control $\sum_{k\geq1}k^2/3^k=3/2$ was checked first. Inversion and conjugation were recomputed at all 494 inputs. All 78 full torsion distribution identities for the thirteen orders and $1\leq M\leq6$ were checked. The square-lattice origin agrees with a separate Hurwitz-zeta computation of the Dirichlet beta value. Six direct lattice sums with explicit coarse tail bounds additionally check the Eisenstein-Kronecker normalization; these use floating arithmetic with a roundoff allowance and do not certify the recorded precision. The attached checks.py and numerical-evidence.json give the methods and measurements.



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