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Bloch elliptic dilogarithm at CM torsion points

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Numbers
$\Delta$
$N$
$r$
$s$ 
$D_q(u)$
-3
3
0
1:
0.6285679457343192761878297399981355823065475038999317288301198552460873906213873237926952674398809899
-3
3
1
0:
-0.3142839728671596380939148699990677911532737519499658644150599276230436953106936618963476337199404950
-3
3
1
1:
0.3142839728671596380939148699990677911532737519499658644150599276230436953106936618963476337199404950
-3
4
0
1:
0.8605828136540144020612912932527080301486360032187327735933559816268238820711968243980178853538275614
-3
4
1
0:
-0.4302914068270072010306456466263540150743180016093663867966779908134119410355984121990089426769137807
-3
4
1
1:
0.4302914068270072010306456466263540150743180016093663867966779908134119410355984121990089426769137807
-3
4
1
2:
0.3066559315871221954380730907959251811135433516315994711733942017587830001133166576192332036924086519
-3
4
1
3:
-0.3066559315871221954380730907959251811135433516315994711733942017587830001133166576192332036924086519
-3
5
0
1:
0.9447480043306547955375208675459788991113896576375703264188914565413178864489800970805352198999195939
-3
5
0
2:
0.3924180442035204751345503191335404894932379128473528457013865167141119581924732643183602806117365488
-3
5
1
0:
-0.4723740021653273977687604337729894495556948288187851632094457282706589432244900485402676099499597969
-3
5
1
1:
0.4723740021653273977687604337729894495556948288187851632094457282706589432244900485402676099499597969
-3
5
1
2:
0.5336246766302560207274549067654101775042544231735224482705327993970820391437138029705369705151843483
-3
5
1
4:
-0.5336246766302560207274549067654101775042544231735224482705327993970820391437138029705369705151843483
-3
5
2
0:
-0.1962090221017602375672751595667702447466189564236764228506932583570559790962366321591801403058682744
-3
5
2
2:
0.1962090221017602375672751595667702447466189564236764228506932583570559790962366321591801403058682744
-3
5
2
3:
0.09653673942108815706277907792935641539260369997292741850368570692532747307007506310335674827477240968
-3
5
2
4:
-0.09653673942108815706277907792935641539260369997292741850368570692532747307007506310335674827477240968
-3
6
0
1:
0.9670750659458414014282334909757396369114169621085995402470898059765535454399128553346037422122363829
-3
6
1
0:
-0.4835375329729207007141167454878698184557084810542997701235449029882767727199564276673018711061181915
-3
6
1
1:
0.4835375329729207007141167454878698184557084810542997701235449029882767727199564276673018711061181915
-3
6
1
2:
0.6732347251343150882980513837007713729899550230319494200301433600018176862135613463150296765067525222
-3
6
1
3:
0.2712375316003302481880107247435474925181879931730009369477854612080487737360185803850723994634081331
-3
6
1
4:
-0.2712375316003302481880107247435474925181879931730009369477854612080487737360185803850723994634081331
-3
6
1
5:
-0.6732347251343150882980513837007713729899550230319494200301433600018176862135613463150296765067525222
-3
6
2
3:
0.2591259579281490046147662842542793607923901400436499634468304496543375439821026455622919789372006841
-3
6
2
5:
-0.2591259579281490046147662842542793607923901400436499634468304496543375439821026455622919789372006841
-3
6
3
1:
-0.01211157367218124357324444048926813172579785312935097350095501155371122975391593482278042052620744903
-3
6
3
2:
0.01211157367218124357324444048926813172579785312935097350095501155371122975391593482278042052620744903
-4
3
0
1:
0.7002469245278938954644948424978381633076692059600250103013924955853340739228766950692132013901061100
-4
3
1
1:
0.3829084510897718184538540560467736118979331274455747893616799502909585081481200473876231830441205124
-4
3
1
2:
-0.3829084510897718184538540560467736118979331274455747893616799502909585081481200473876231830441205124
-4
4
0
1:
0.9432623087139065947402392348124305456809454499864044314840291987983217642137583584636771966329804319
-4
4
1
1:
0.5838125667061079904157859645607030272009389163827114604745979333168567257587510242867152116266147896
-4
4
1
3:
-0.5838125667061079904157859645607030272009389163827114604745979333168567257587510242867152116266147896
-4
4
2
1:
0.3594497420077986043244532702517275184800065336036929710094312654814650384550073341769619850063656423
-4
5
0
1:
1.023329324353262993143884266703334518075981019698631622118632628587410569505810596002731705010677380
-4
5
0
2:
0.4410999751556337634122357603427244387549055828413217645794035707773898780782033454915842507177581344
-4
5
1
1:
0.6853962164799013390465602623561439908225031740513393528712033615722665055692602863063295782001264366
-4
5
1
2:
0.3459607450737100641466987529214525952925346211910044352537572283506429560209888684446001944600768724
-4
5
1
3:
-0.3459607450737100641466987529214525952925346211910044352537572283506429560209888684446001944600768724
-4
5
1
4:
-0.6853962164799013390465602623561439908225031740513393528712033615722665055692602863063295782001264366
-4
5
2
1:
0.3832502713073683534764320852837943144018126162236103462086517409506344109900740171914195931732836286
-4
5
2
2:
0.2230619894913061064033406667996019690364331472648360093495612008658829566908489095643220640682134034
-4
5
2
3:
-0.2230619894913061064033406667996019690364331472648360093495612008658829566908489095643220640682134034
-4
5
2
4:
-0.3832502713073683534764320852837943144018126162236103462086517409506344109900740171914195931732836286
-4
6
0
1:
1.038601765700703493200243688118610694762504446185234717417641294955927271792972035042090531314303411
-4
6
1
1:
0.7330319133537187661861014772956926935517677304255872945123761980836255451095583949222297837391735674
-4
6
1
2:
0.5763504273662624396022327532215875569446827675561202911869045198449417339227134638325885558793262144
-4
6
1
4:
-0.5763504273662624396022327532215875569446827675561202911869045198449417339227134638325885558793262144
-4
6
1
5:
-0.7330319133537187661861014772956926935517677304255872945123761980836255451095583949222297837391735674
-4
6
2
1:
0.4176811906472014010969123599960552812398147282988951807170482471977539510353351399917935467063334155
-4
6
2
5:
-0.4176811906472014010969123599960552812398147282988951807170482471977539510353351399917935467063334155
-4
6
3
1:
0.3173384734381220770106407864510645514097360785144502209397125452943755657747566476815900183459855977
-4
6
3
2:
0.3055698523469847270141422108229180012107367157596474229052650968723017266834136401198607475751298440
-7
3
0
1:
0.6726681879614967703089623511470645526925948694713178466547297321917664669716070087926741143105907456
-7
3
1
0:
-0.1877118666230770171095826775818828955671973696426262148789123463867499777544454554411388062961095173
-7
3
1
1:
0.1877118666230770171095826775818828955671973696426262148789123463867499777544454554411388062961095173
-7
4
0
1:
0.9113940316117034983468570180922324031794882161129514112719540688149664615178375257491496677148202420
-7
4
1
0:
-0.2834969624719382988488409790147481519202979745063974097824536356383158023076938688565130612109926233
-7
4
1
1:
0.2834969624719382988488409790147481519202979745063974097824536356383158023076938688565130612109926233
-7
4
1
2:
0.2430742939518980250367977747850550876495206271766776482991368076787463790281483612321900379491656535
-7
4
1
3:
-0.2430742939518980250367977747850550876495206271766776482991368076787463790281483612321900379491656535
-7
5
0
1:
0.9930071893817360252133438495610730517606214142488296733807782188195910921694965758009840556496917442
-7
5
0
2:
0.4223902197694298636993841602472603676220908167476986951688753635293449181747782657965928427803147316
-7
5
1
0:
-0.3324006315286151053095894949450140986479017231865983869328511469245818098943084881244732919057310812
-7
5
1
1:
0.3324006315286151053095894949450140986479017231865983869328511469245818098943084881244732919057310812
-7
5
1
2:
0.4454409205868873709697957081549111545906775763614983266102441496855253943198650981192494768346876674
-7
5
1
4:
-0.4454409205868873709697957081549111545906775763614983266102441496855253943198650981192494768346876674
-7
5
2
0:
-0.1105438562887668429623955681841552826273891563685074150493554421258310616613003543416096589660830782
-7
5
2
2:
0.1105438562887668429623955681841552826273891563685074150493554421258310616613003543416096589660830782
-7
5
2
3:
0.06497165638481286667491347264056910387932280880458348822930662713248850194838467557138462115635141300
-7
5
2
4:
-0.06497165638481286667491347264056910387932280880458348822930662713248850194838467557138462115635141300
-7
6
0
1:
1.010982977273282389021125527169790824425322137472601618426548609342331378104998877817441679694569758
-7
6
1
0:
-0.3558789136134512096868232653686490337403460870104556765425947794346915944973472076393073348737572037
-7
6
1
1:
0.3558789136134512096868232653686490337403460870104556765425947794346915944973472076393073348737572037
-7
6
1
2:
0.5798026023154768785330120125807201026022111012828171634375005645257323169181784633863199652768778066
-7
6
1
3:
0.2508548433922921704798515623779637492825387927053867950871773922841886228505663669975965363533498942
-7
6
1
4:
-0.2508548433922921704798515623779637492825387927053867950871773922841886228505663669975965363533498942
-7
6
1
5:
-0.5798026023154768785330120125807201026022111012828171634375005645257323169181784633863199652768778066
-7
6
2
3:
0.1765437297134564864574023133821390588929887601588704408629511324296219949848873425199974109237569665
-7
6
2
5:
-0.1765437297134564864574023133821390588929887601588704408629511324296219949848873425199974109237569665
-7
6
3
1:
-0.0009903476655186167788410002245969976932149166328124242222270055273408388237941823142152541143418196278
-7
6
3
2:
0.0009903476655186167788410002245969976932149166328124242222270055273408388237941823142152541143418196278
-8
3
0
1:
0.6789970233458784073465906534227935765523913799860153051358260428968702344721848496683677813301919405
-8
3
1
1:
0.1895548306801968176133067958793559558206097927211492732023212623089918084592468038166103332413897598
-8
3
1
2:
-0.1895548306801968176133067958793559558206097927211492732023212623089918084592468038166103332413897598
-8
4
0
1:
0.9187015893151005531163048157960779969460127134376734839387977079594304267888311203478017767705111198
-8
4
1
1:
0.3582561789432132606410066527391631811681457325564171892238164644263189503215357402611546575268149227
-8
4
1
3:
-0.3582561789432132606410066527391631811681457325564171892238164644263189503215357402611546575268149227
-8
4
2
1:
0.1280796271349248796610240287708653394928383627838364833845843509804082686557610726340314606007813790
-8
5
0
1:
0.9999568829957085015969729131628307928100636926727021142156881401448669968986661885313480077434239516
-8
5
0
2:
0.4266858291754867763154411431213592787470365649318084724346466180793706431528692752571707935921734586
-8
5
1
1:
0.4700075908863703812701592015562886870768484707290215320613408806101684389889467191456060001161758415
-8
5
1
2:
0.2521478662503990967451032650975718798424469203216027627231784136495451336964974900308012602979928026
-8
5
1
3:
-0.2521478662503990967451032650975718798424469203216027627231784136495451336964974900308012602979928026
-8
5
1
4:
-0.4700075908863703812701592015562886870768484707290215320613408806101684389889467191456060001161758415
-8
5
2
1:
0.1540662677350452921581225372532743505429447651256891860480200983815377238945803005358986431607920687
-8
5
2
2:
0.09298914838576640989564611713558133775089735454853642243816834210303207576252809303456926837368467661
-8
5
2
3:
-0.09298914838576640989564611713558133775089735454853642243816834210303207576252809303456926837368467661
-8
5
2
4:
-0.1540662677350452921581225372532743505429447651256891860480200983815377238945803005358986431607920687
-8
6
0
1:
1.017311203286469670652314323285969182860176722593734653160738582216369069672912895739356626592874651
-8
6
1
1:
0.5365589582826782214288814398103004580621485484561401519020713944125317356942355899355833400081979535
-8
6
1
2:
0.4498310429782271581799655924961224973667576918155751380967752774183299263305368630527206722711186313
-8
6
1
4:
-0.4498310429782271581799655924961224973667576918155751380967752774183299263305368630527206722711186313
-8
6
1
5:
-0.5365589582826782214288814398103004580621485484561401519020713944125317356942355899355833400081979535
-8
6
2
1:
0.1976043307158441631710443465048559730355238324411588959981857764692859033251714788420528321250053176
-8
6
2
5:
-0.1976043307158441631710443465048559730355238324411588959981857764692859033251714788420528321250053176
-8
6
3
1:
0.1115168630376358948742419226329468910589723492078557900718627380988310911298088169817318270084359205
-8
6
3
2:
0.1103325313052879545066702657847257090905620018225674855288622559698948090944444382185367816060226611
-11
3
0
1:
0.6760374494336952120354976067099661371559495986576046176502297510113064564546533088328662344463549331
-11
3
1
0:
-0.1151876129423552601325003102505817980501199782237654691475933594916119248662363191795030149912948803
-11
3
1
1:
0.1151876129423552601325003102505817980501199782237654691475933594916119248662363191795030149912948803
-11
4
0
1:
0.9152839978329721938161743242977902027252940783814856907481441821925955406177854593379742024063548394
-11
4
1
0:
-0.1946274810515036507995879547004056030372455589797092209272926496934314612274267653858783450661791150
-11
4
1
1:
0.1946274810515036507995879547004056030372455589797092209272926496934314612274267653858783450661791150
-11
4
1
2:
0.1776641085615520088168883110023897096943921325402286173721180241592907806198494511671052959209296133
-11
4
1
3:
-0.1776641085615520088168883110023897096943921325402286173721180241592907806198494511671052959209296133
-11
5
0
1:
0.9967064604707158330886114256199895823436272480785855743619161266580573479584124640531154774215362686
-11
5
0
2:
0.4246771808714900213039891240155960084386140249359076836277018763750125853058789322431255763764622790
-11
5
1
0:
-0.2448176942998108172821172567501234034617009736714829118511576329279725316042316786730002721465639446
-11
5
1
1:
0.2448176942998108172821172567501234034617009736714829118511576329279725316042316786730002721465639446
-11
5
1
2:
0.3502721381393859681509078783749788056079778725196577606540882204509916474321543183857850930044305458
-11
5
1
4:
-0.3502721381393859681509078783749788056079778725196577606540882204509916474321543183857850930044305458
-11
5
2
0:
-0.06314925955415437429530492319379172834662710750368497550267254185535024374678556594057899008584080258
-11
5
2
2:
0.06314925955415437429530492319379172834662710750368497550267254185535024374678556594057899008584080258
-11
5
2
3:
0.03828353834510047856534544769551653321858495733671611286062697349134518136346191113390205769765455494
-11
5
2
4:
-0.03828353834510047856534544769551653321858495733671611286062697349134518136346191113390205769765455494
-11
6
0
1:
1.014351335083575382457291574849193186623735225723776166452595323011095975611357168475780055711850199
-11
6
1
0:
-0.2743864494849217761639238110449883915636741532856332210136200697494309026690514313867833832707919423
-11
6
1
1:
0.2743864494849217761639238110449883915636741532856332210136200697494309026690514313867833832707919423
-11
6
1
2:
0.4765559238494118855041066332953813837222604458123442083699081079423191722589987437662846920714733715
-11
6
1
3:
0.2134774854871254715520008645497161068611645560746668797734241700718571441166341809984720888152699530
-11
6
1
4:
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0.00001943481158305119658085491593124021802970642794636893659860163239054713647118699620430548275120892125
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Definition
Bloch's elliptic dilogarithm at the torsion point $u=(r\tau_\Delta+s)/N$ of the marked CM torus is $D_q(u)=\sum_{n\in\mathbb Z}D(e^{2\pi i u}q^n)$, where $q=e^{2\pi i\tau_\Delta}$ and $D$ is the Bloch-Wigner dilogarithm [3].
Parameters
$\Delta$
—   CM order discriminant ($\Delta<0$ is an imaginary quadratic order discriminant)
$N$
—   exact order of the torsion point ($N\geq3$)
$r$
—   $\tau_\Delta$ coordinate numerator ($0\leq r<N$; together with $s$, it satisfies (5))
$s$
—   real coordinate numerator ($0\leq s<N$; together with $r$, it satisfies (5))
Formulas
(1)
$D(z)=\operatorname{Im}\bigl(\operatorname{Li}_2(z)+\log|z|\log(1-z)\bigr)$, continuously extended by $D(0)=D(1)=0$ [1].
(2)
$D_q(u)=D(x)+\sum_{n=1}^{\infty}\bigl(D(xq^n)-D(q^n/x)\bigr)$ for $x=e^{2\pi i u}$ with $|q|<|x|\leq1$. This follows from $D(1/z)=-D(z)$ [1].
(3)
$D_q(-u)=-D_q(u)$. For real $q$, complex conjugation of $x$ negates the value; in the stated coordinates, $D_q((r\tau_\Delta+(br-s))/N)=-D_q((r\tau_\Delta+s)/N)$. Coordinates are interpreted modulo $N$.
Comments
(4)
Put $b=0$ when $4\mid\Delta$ and $b=1$ otherwise, $\tau_\Delta=(-b+i\sqrt{|\Delta|})/2$, and $E_\Delta=\mathbb C/(\mathbb Z+\tau_\Delta\mathbb Z)$. The square root is positive, and $u=(r\tau_\Delta+s)/N$. The marking is part of the convention: it is not replaced by a modularly equivalent representative. In particular, $q=e^{2\pi i\tau_\Delta}$ is negative for odd $\Delta$.
(5)
The condition $\gcd(N,r,s)=1$ makes $N$ the exact order of the point. Among $(r,s)$ and $((-r)\bmod N,(-s)\bmod N)$, the lexicographically smaller pair is listed. The condition $2s-br\not\equiv0\pmod N$ removes the real-$x$ locus, where every summand vanishes. The origin and all points of order two also vanish. Omitted opposite points have the negative of the listed value; signs are retained.
(6)
All logarithms are natural. The principal dilogarithm and principal logarithm in $D$ give the standard single-valued Bloch-Wigner function. Its continuous value is zero on the real axis. Classical polylogarithm values appear in the table of classical polylogarithm values.
(7)
The current discriminants are exactly the thirteen CM orders with rational singular modulus, including nonmaximal orders [4]. Thus $\Delta$ is an order discriminant and need not be a fundamental field discriminant. At these orders, the principal lattice represents the unique ideal class.
(8)
On the unit circle, $D(e^{i\theta})=\operatorname{Cl}_2(\theta)$, the classical Clausen function. The elliptic average has additional nonzero terms. Guillera and Rogers connect the points $u=1/3,1/4,1/6$ to Hessian Mahler measures and reciprocal Laurent Mahler measures by theta parametrizations [1], Theorem 4.
(9)
For the Eisenstein-Kronecker phase $e^{2\pi i(n\xi-m\eta)}$ at $u=\xi\tau+\eta$, this definition follows the negative sign in [3], Proposition 6. The positive sign printed with that phase in the introductory regulator formula of [2] is inconsistent with the defining $D$-series. This distinction does not change the shared $D$-series definition.
Programs
(P1)
Sage
# Download the attached generate.py into this directory first.
from generate import Table
g = Table()
print(g.value({'Delta': -4, 'N': 4, 'r': 0, 's': 1}, 100))
References
[1]
Jesús Guillera and Mathew Rogers, Mahler measure and the WZ algorithm (2013), section 3, Theorem 4. (arXiv)
[2]
Detchat Samart, The elliptic trilogarithm and Mahler measures of K3 surfaces (2014), section 1. (arXiv)
Links
Similar tables
Rational singular moduli —   gives the rational singular moduli of the same thirteen CM orders
Clausen functions at rational multiples of pi —   gives the classical unit-circle values $D(e^{i\theta})$ used in the elliptic average
Polylogarithm values at rational arguments —   gives classical polylogarithm values, including the weight-two function used to define $D$
Mahler measures of the reciprocal Laurent family —   contains the reciprocal Laurent family connected to elliptic dilogarithms at points of order four by [1], Theorem 4
Mahler measures of the Hessian family —   contains ordinary Hessian Mahler measures connected to elliptic dilogarithms at points of order three and six by [1], Theorem 4
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds the 362 signed 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 $3\leq N\leq6$, with all torsion directions modulo inversion except the symmetry-forced zeros specified in (5). This arithmetic reference covers rectangular and rhombic CM tori, the orders $3,4,6$ occurring in Mahler-measure identities, the next prime order $5$, and the first mixed-prime order $6$. The cutoff six and the representative convention 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 finite coverage)
How they were obtained:

Each value is enclosed by Arb ball evaluation of the paired Bloch-Wigner series, including uncertainty in the CM argument. For $|z|=t<1$, the bound $|D(z)|\leq t(1+|\log t|)/(1-t)$ gives an explicit geometric tail bound. The generator adds that bound to the computed ball; the enclosures justify the 100 significant digits recorded. It uses 128 extra working bits, and the least relative accuracy measured over the selection was 421 bits. Between 7 and 57 terms on each side were needed.

more

Every entry was recomputed with 30 extra decimal working digits and compared with an independent Fourier expansion whose unit-circle term is a finite Hurwitz-zeta sum. Relative differences were below $5.2\times10^{-128}$. Point inversion and real-$q$ conjugation identities were checked for all entries, and the 132 omitted inversion representatives were checked as zeros. A separate ball integral for the reciprocal Laurent Mahler measure at the square-lattice point $u=1/4$ agrees with [1], equation (26). Its nonzero integration control was $\int_0^1t^2\,dt=1/3$. A directly truncated Eisenstein-Kronecker sum with an explicit tail bound determines the negative lattice sign; this last check establishes the sign, not the recorded precision. The attached checks.py and numerical-evidence.json give the methods and measurements.



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