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Elliptic dilogarithm companion J at CM torsion points

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Numbers
$\Delta$
$N$
$r$
$s$ 
$J_q(u)$
-3
3
1
0:
0.5443558090105189708421163475716256353542699034879485658280493090067524417908638576633704729926895505
-3
3
1
1:
0.5443558090105189708421163475716256353542699034879485658280493090067524417908638576633704729926895505
-3
3
1
2:
0
comment: The order-three automorphism fixes this point modulo the lattice, so the value is exactly zero, as in (10).
-3
4
1
0:
0.7452865786846661428838062280963537053831906739083116106866543253039144822018440641962107542251952272
-3
4
1
1:
0.7452865786846661428838062280963537053831906739083116106866543253039144822018440641962107542251952272
-3
4
1
2:
0.1770478846504204619405153442852657126467300202220344377064896514412358921027645154213852356663412116
-3
4
1
3:
0.1770478846504204619405153442852657126467300202220344377064896514412358921027645154213852356663412116
-3
4
2
1:
0.3540957693008409238810306885705314252934600404440688754129793028824717842055290308427704713326824232
-3
5
1
0:
0.8181757719249979106091310364721920740135152740600562826019408267035715233716665162706174452106292733
-3
5
1
1:
0.8181757719249979106091310364721920740135152740600562826019408267035715233716665162706174452106292733
-3
5
1
2:
0.3080883506987053143728252629044692073414946551004046746002513722379181500759905372720474204503740929
-3
5
1
3:
0.1114710249829080095377081463486992911533050643700459305884292409675812048461876433265046250188595843
-3
5
1
4:
0.3080883506987053143728252629044692073414946551004046746002513722379181500759905372720474204503740929
-3
5
2
0:
0.3398439951836535130578269222790048603477568488586309679465458779871804355062398116383490006648117258
-3
5
2
1:
0.6161767013974106287456505258089384146829893102008093492005027444758363001519810745440948409007481858
-3
5
2
2:
0.3398439951836535130578269222790048603477568488586309679465458779871804355062398116383490006648117258
-3
5
2
3:
-0.05573551249145400476885407317434964557665253218502296529421462048379060242309382166325231250942979216
-3
5
2
4:
-0.05573551249145400476885407317434964557665253218502296529421462048379060242309382166325231250942979216
-3
6
1
0:
0.8375115744756099318954648585877305808117758147827421250755395889239158057633671313162108556700608324
-3
6
1
1:
0.8375115744756099318954648585877305808117758147827421250755395889239158057633671313162108556700608324
-3
6
1
2:
0.3886922497841005268073398590844150740733442461295411169506091276741773080586534809952146640292338885
-3
6
1
3:
0.1426138212438926431519448874243411263193535363674605772638216119453171182092141018332803484866001458
-3
6
1
4:
0.1426138212438926431519448874243411263193535363674605772638216119453171182092141018332803484866001458
-3
6
1
5:
0.3886922497841005268073398590844150740733442461295411169506091276741773080586534809952146640292338885
-3
6
2
1:
0.7773844995682010536146797181688301481466884922590822339012182553483546161173069619904293280584677770
-3
6
2
3:
0.1635916822037241187842352246546332540997244959182798535972872373591042612862854466544354946676266524
-3
6
2
5:
0.1635916822037241187842352246546332540997244959182798535972872373591042612862854466544354946676266524
-3
6
3
1:
0.3062055034476167619361801120789743804190780322857404308611088493044213794954995484877158431542267981
-3
6
3
2:
0.3062055034476167619361801120789743804190780322857404308611088493044213794954995484877158431542267981
-4
3
1
0:
0.7002469245278938954644948424978381633076692059600250103013924955853340739228766950692132013901061100
-4
3
1
1:
0.3829084510897718184538540560467736118979331274455747893616799502909585081481200473876231830441205124
-4
3
1
2:
0.3829084510897718184538540560467736118979331274455747893616799502909585081481200473876231830441205124
-4
4
1
0:
0.9432623087139065947402392348124305456809454499864044314840291987983217642137583584636771966329804319
comment: Square-lattice rotation gives the Bloch elliptic dilogarithm at $u=1/4$.
-4
4
1
1:
0.5838125667061079904157859645607030272009389163827114604745979333168567257587510242867152116266147896
-4
4
1
2:
0.3594497420077986043244532702517275184800065336036929710094312654814650384550073341769619850063656423
-4
4
1
3:
0.5838125667061079904157859645607030272009389163827114604745979333168567257587510242867152116266147896
-4
5
1
0:
1.023329324353262993143884266703334518075981019698631622118632628587410569505810596002731705010677380
-4
5
1
1:
0.6853962164799013390465602623561439908225031740513393528712033615722665055692602863063295782001264366
-4
5
1
2:
0.3832502713073683534764320852837943144018126162236103462086517409506344109900740171914195931732836286
-4
5
1
3:
0.3832502713073683534764320852837943144018126162236103462086517409506344109900740171914195931732836286
-4
5
1
4:
0.6853962164799013390465602623561439908225031740513393528712033615722665055692602863063295782001264366
-4
5
2
0:
0.4410999751556337634122357603427244387549055828413217645794035707773898780782033454915842507177581344
-4
5
2
1:
0.3459607450737100641466987529214525952925346211910044352537572283506429560209888684446001944600768724
-4
5
2
2:
0.2230619894913061064033406667996019690364331472648360093495612008658829566908489095643220640682134034
-4
5
2
3:
0.2230619894913061064033406667996019690364331472648360093495612008658829566908489095643220640682134034
-4
5
2
4:
0.3459607450737100641466987529214525952925346211910044352537572283506429560209888684446001944600768724
-4
6
1
0:
1.038601765700703493200243688118610694762504446185234717417641294955927271792972035042090531314303411
-4
6
1
1:
0.7330319133537187661861014772956926935517677304255872945123761980836255451095583949222297837391735674
-4
6
1
2:
0.4176811906472014010969123599960552812398147282988951807170482471977539510353351399917935467063334155
-4
6
1
3:
0.3173384734381220770106407864510645514097360785144502209397125452943755657747566476815900183459855977
-4
6
1
4:
0.4176811906472014010969123599960552812398147282988951807170482471977539510353351399917935467063334155
-4
6
1
5:
0.7330319133537187661861014772956926935517677304255872945123761980836255451095583949222297837391735674
-4
6
2
1:
0.5763504273662624396022327532215875569446827675561202911869045198449417339227134638325885558793262144
-4
6
2
3:
0.3055698523469847270141422108229180012107367157596474229052650968723017266834136401198607475751298440
-4
6
2
5:
0.5763504273662624396022327532215875569446827675561202911869045198449417339227134638325885558793262144
-7
3
1
0:
0.9475275568799607284126186551255398027921706341070349493027496614068959455646114364761375281852042046
-7
3
1
1:
0.9475275568799607284126186551255398027921706341070349493027496614068959455646114364761375281852042046
-7
3
1
2:
0.6630505417213983662788537975895574180844536733320975672373137087251864253366982750075626079119904659
-7
4
1
0:
1.270746479427067145485974026035503429656423552711967946730423583896483023773152348835894561390545259
-7
4
1
1:
1.270746479427067145485974026035503429656423552711967946730423583896483023773152348835894561390545259
-7
4
1
2:
0.8879743571597869398164722296109194974093488185360036321512423199838280639285038617128755380821206273
-7
4
1
3:
0.8879743571597869398164722296109194974093488185360036321512423199838280639285038617128755380821206273
-7
4
2
1:
0.1301710048210268385960165798314161496659376919959446557931507358770684078027207998280907659215766338
-7
5
1
0:
1.374633836003030179211422408935690676793163456320315486310885266810384888506164425505672286900499283
-7
5
1
1:
1.374633836003030179211422408935690676793163456320315486310885266810384888506164425505672286900499283
-7
5
1
2:
0.9843959312491548759111055734729077934752488474220987344026023794184033346632775393902320289687310509
-7
5
1
3:
0.8085107281894708793382584499485336235514874848404269484603850587622865099736387237652936397432251706
-7
5
1
4:
0.9843959312491548759111055734729077934752488474220987344026023794184033346632775393902320289687310509
-7
5
2
0:
0.5984563023600608359353130298993735448975744931922661663057183166704618830164561458611528906331125393
-7
5
2
1:
0.7083304967187414326741286967605896958626055974220177156423375835182395749826104510286552253210745604
-7
5
2
2:
0.5984563023600608359353130298993735448975744931922661663057183166704618830164561458611528906331125393
-7
5
2
3:
0.4291231657300459887906851767132656742516657037758974226255147432207146389029413601020148466660002414
-7
5
2
4:
0.4291231657300459887906851767132656742516657037758974226255147432207146389029413601020148466660002414
-7
6
1
0:
1.392455741932603861672440642457489963862183331915114564087399113816877212429798050748989996053516390
-7
6
1
1:
1.392455741932603861672440642457489963862183331915114564087399113816877212429798050748989996053516390
-7
6
1
2:
1.024092160246485443677256681738160402151730062609383455185647754329858175871475200612126055005925400
-7
6
1
3:
0.7817649214496111368429052057281504482710572157740105574934212342255993392304368677242468762072065882
-7
6
1
4:
0.7817649214496111368429052057281504482710572157740105574934212342255993392304368677242468762072065882
-7
6
1
5:
1.024092160246485443677256681738160402151730062609383455185647754329858175871475200612126055005925400
-7
6
2
1:
1.053608507910873337936232667091984677176779615220620559515324945571936713737902988712908198143865101
-7
6
2
3:
0.7529293280622739058964178654973307079449845965285726976266958559321326333133177637590305799829166717
-7
6
2
5:
0.7529293280622739058964178654973307079449845965285726976266958559321326333133177637590305799829166717
-7
6
3
1:
0.1127221599148932868079064434141669173288089277762315867030407850987762222475287926335053169986671682
-7
6
3
2:
0.1127221599148932868079064434141669173288089277762315867030407850987762222475287926335053169986671682
-8
3
1
0:
1.116280561369594596764273186548790534455537507911351000529305112372260119926162194002733728894732996
-8
3
1
1:
0.9042281741175134458485055941845996947298589140152278422776375627840042197368593274684021401723283341
-8
3
1
2:
0.9042281741175134458485055941845996947298589140152278422776375627840042197368593274684021401723283341
-8
4
1
0:
1.480372193141032351314731104506136270596038911530928594570559049777832423883498440529641163539399414
-8
4
1
1:
1.220717929013547971945384024104044372517389073342352853939760238846770133044419762830073510912346105
-8
4
1
2:
1.013301494130909255246154202852535693908729045331869837925941304239832143986215710066918083262374100
-8
4
1
3:
1.220717929013547971945384024104044372517389073342352853939760238846770133044419762830073510912346105
-8
5
1
0:
1.586990331234421638980094024550148795258659454069975253967703368929879715173244983732238677754202294
-8
5
1
1:
1.331749931566659863428105548074085120025285410897023712778668425201931642370742523068515065044538289
-8
5
1
2:
1.033151249073524348936726554664663819809910788508231389486631164844653260109723733088584739110052484
-8
5
1
3:
1.033151249073524348936726554664663819809910788508231389486631164844653260109723733088584739110052484
-8
5
1
4:
1.331749931566659863428105548074085120025285410897023712778668425201931642370742523068515065044538289
-8
5
2
0:
0.7089830762152246893144818941869052519997991399328968987470242799413821386775974475500319020889353110
-8
5
2
1:
0.6539640867982020187450677908789606147312927285545467309300390619733363486444607210294823219619515509
-8
5
2
2:
0.5706811132745619331175066564828346850818088339664448611313583474240056072373387200472754876390362356
-8
5
2
3:
0.5706811132745619331175066564828346850818088339664448611313583474240056072373387200472754876390362356
-8
5
2
4:
0.6539640867982020187450677908789606147312927285545467309300390619733363486444607210294823219619515509
-8
6
1
0:
1.596403960982945740781243596005008549734039234134332526303607575979752539284415788402472691622252007
-8
6
1
1:
1.359456386631647826115831316226484919688732525066255780596038410404677222050751177078578534488876953
-8
6
1
2:
1.038263680298148442399107002730613282506270480178657026038955019793622479321334691401105827174885509
-8
6
1
3:
0.9191788765531373991194764214199814993831600357161795043281989020958962593596893418674445808933242123
-8
6
1
4:
1.038263680298148442399107002730613282506270480178657026038955019793622479321334691401105827174885509
-8
6
1
5:
1.359456386631647826115831316226484919688732525066255780596038410404677222050751177078578534488876953
-8
6
2
1:
1.041377805753526099742179927680198660100214634222071043218537086022293371766796877277081151405269960
-8
6
2
3:
0.8411619954816912447543102376018042474338930079834855298378488095172586187548618392658166791734767253
-8
6
2
5:
1.041377805753526099742179927680198660100214634222071043218537086022293371766796877277081151405269960
-11
3
1
0:
1.396649622466814676309632686777658106814833566826817580900026200380657207504331640853957906972339943
-11
3
1
1:
1.396649622466814676309632686777658106814833566826817580900026200380657207504331640853957906972339943
-11
3
1
2:
1.227547034160091889758355830048256044358380759821817381257879784057609665450141834411982150521695248
-11
4
1
0:
1.834422248700280504871464762869753700207593163243933101173787269115442803701071239115404783108305854
-11
4
1
1:
1.834422248700280504871464762869753700207593163243933101173787269115442803701071239115404783108305854
-11
4
1
2:
1.558215393210491996404034831061193481979471027308159314388627442033461732807821311668699106890130787
-11
4
1
3:
1.558215393210491996404034831061193481979471027308159314388627442033461732807821311668699106890130787
-11
4
2
1:
0.05691797391350512230602285111023486476456219090230565297151701578332030702694254055502638989777870839
-11
5
1
0:
1.952889646730029280613861754387246158947739526839375108160371866193997348241979604878515317953549069
-11
5
1
1:
1.952889646730029280613861754387246158947739526839375108160371866193997348241979604878515317953549069
-11
5
1
2:
1.644365465806560677178155142997128772476953260242691395956580211713237121683149968233789839494960151
-11
5
1
3:
1.490679454343378226687602653563471926661544326149757997108031994960888015415432972243520380801551008
-11
5
1
4:
1.644365465806560677178155142997128772476953260242691395956580211713237121683149968233789839494960151
-11
5
2
0:
0.8917920267321799388654383386676951730687055675413493051810739502625649727165694631075122739356317897
-11
5
2
1:
0.9456031898973275965448855613767206352290113535381214461164968395192184971211024003419043829281192120
-11
5
2
2:
0.8917920267321799388654383386676951730687055675413493051810739502625649727165694631075122739356317897
-11
5
2
3:
0.8067193484151092238534903627556432196184226625494379663342274941443132830545755328746098227201663100
-11
5
2
4:
0.8067193484151092238534903627556432196184226625494379663342274941443132830545755328746098227201663100
-11
6
1
0:
1.954684868895361336495388510387377176805348930570589547860605993977299272566966727654297486356237951
-11
6
1
1:
1.954684868895361336495388510387377176805348930570589547860605993977299272566966727654297486356237951
-11
6
1
2:
1.648871874925806822603173984386992161585592169592694698907075062382952981882304390871053845295637750
-11
6
1
3:
1.422604768683658131919014013075015679718834979675380989696679324208713296481523768209238308424933364
-11
6
1
4:
1.422604768683658131919014013075015679718834979675380989696679324208713296481523768209238308424933364
-11
6
1
5:
1.648871874925806822603173984386992161585592169592694698907075062382952981882304390871053845295637750
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6
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24.83355708914683790301548873267183112258288210873275765683031119263569112243881099557726731251634334
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6.779204050728038589082166160289460102047571126323828883385190240819550760977653334341815138800998643e-8
Definition
The real elliptic dilogarithm companion $J_q(u)$ is the logarithmic elliptic average with Bernoulli correction in (1), evaluated at the torsion point $u=(r\tau_\Delta+s)/N$ of the marked CM torus [3], equation (14).
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 (6))
$s$
—   real coordinate numerator ($0\leq s<N$; together with $r$, it satisfies (6))
Formulas
(1)
$J_q(u)=\sum_{n=0}^{\infty}j(xq^n)-\sum_{n=1}^{\infty}j(q^n/x)+\frac{\log^2|q|}{3}B_3\!\left(\frac{\log|x|}{\log|q|}\right)$, where $x=e^{2\pi i u}$, $j(z)=\log|z|\log|1-z|$, $j(1)=0$, and $B_3(t)=t^3-\frac32t^2+\frac12t$ [3].
(2)
$J_q(-u)=-J_q(u)$. For real $q$, conjugating $x$ preserves the value: $J_q((r\tau_\Delta+(br-s))/N)=J_q((r\tau_\Delta+s)/N)$. Coordinates are interpreted modulo $N$.
(3)
For the square marking $\tau=i$ and $q=e^{-2\pi}$, $J_q((ri+s)/N)=D_q((r-si)/N)$, where $D_q$ is the Bloch elliptic dilogarithm. Multiplication of the square lattice by $i$ reindexes the Eisenstein-Kronecker expression in [3], Proposition 6.
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)
All logarithms are natural. Write $x=e^{2\pi i u}$, $j(z)=\log|z|\log|1-z|$ for $z\ne1$, and $j(1)=0$ by continuity. The polynomial $B_3(t)=t^3-\frac32t^2+\frac12t$ is the third Bernoulli polynomial. Since the logarithms act on absolute values, no logarithm branch is selected.
(6)
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 origin and all points of order two vanish by oddness. The locus satisfying both $2r\equiv0\pmod N$ and $br\equiv0\pmod N$ is also omitted: conjugating $x$ then represents the inverse point, so (2) forces zero. Omitted opposite points have the negative of the listed value; signs are retained.
(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)
The Bloch elliptic dilogarithm $D_q$ is the other real component of the complex regulator $D_q-iJ_q$. One component does not determine the other. The two tables use the same marked CM lattices and torsion coordinates, with different symmetry-forced zero loci.
(9)
The defining logarithmic series agrees in [3], equation (14), and [2], section 1. For the Eisenstein-Kronecker phase $e^{2\pi i(n\xi-m\eta)}$ at $u=\xi\tau+\eta$, the regulator convention follows the negative sign in Brunault, Proposition 6. The positive sign printed with that phase in Samart is inconsistent with the defining series.
(10)
At $\Delta=-3$, the point $u=(\tau+2)/3$ satisfies $\tau u=u-1$ because $\tau^2+\tau+1=0$. Multiplication of the lattice by $\tau$ multiplies the regulator by $1/\tau$ in [3], Proposition 6. Periodicity therefore forces both components to vanish. This additional zero is retained exactly.
(11)
Guillera and Rogers connect elliptic dilogarithms at points of orders three, four and six to Hessian Mahler measures and reciprocal Laurent Mahler measures [1], Theorem 4. These identities give arithmetic context for the torsion selection; they do not assert a separate Mahler-measure identity for every listed $J_q$ value.
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': 1, 's': 0}, 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
Bloch elliptic dilogarithm at CM torsion points —   gives the other real regulator component on the same marked CM lattices and torsion groups
Bernoulli polynomials —   contains the polynomial $B_3$ in the logarithmic correction
Rational singular moduli —   gives the rational singular moduli of the same thirteen CM orders
Mahler measures of the reciprocal Laurent family —   contains the Mahler-measure family that supplies a square-lattice control through $D_q(1/4)=J_q(i/4)$
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds 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 every exact order $3\leq N\leq6$, with all torsion directions modulo inversion except the symmetry-forced zeros specified in (6). This arithmetic reference covers rectangular and rhombic CM tori, the orders $3,4,6$ in weight-two torsion identities, the next prime order $5$, and the first mixed-prime order $6$. The cutoff six and lexicographic representatives are noncanonical, notation-independent choices; 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 evaluation of the defining logarithmic series, including uncertainty in the CM argument and the Bernoulli correction. For $|z|<1$, the bound $|\log|1-z||\leq |z|/(1-|z|)$ yields explicit geometric bounds for both omitted tails. The generator adds those bounds to the ball, which justifies the 100 significant digits recorded. It uses 128 extra working bits; apart from the exact zero identified in (10), the minimum measured relative accuracy was 425 bits, with 8 to 57 terms in each half of the series.

more

Every entry was recomputed with 30 extra decimal working digits and compared with a separately implemented Fourier expansion. Inversion and conjugation were checked for every row, and the 132 omitted inversion representatives were checked as zeros. All 26 square-lattice rows were also compared with independently computed Bloch dilogarithms after rotation. A separate ball integral for the reciprocal Laurent Mahler measure agrees at $u=i/4$, with the nonzero control $\int_0^1t^2\,dt=1/3$. A bounded lattice sum distinguishes the negative regulator sign in Brunault from the positive sign in Samart; this establishes the sign, not the 100-digit precision. The attached checks.py and numerical-evidence.json give the computations and measured errors.



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