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Modified Mahler measures of the Hessian family

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Numbers
$t$ 
$\widetilde N(t)$
0:
0
comment: The continuous modified period is exactly $\widetilde N(0)=0$, unlike the ordinary value $N(0)$. This is an intrinsic specialization, not a nonsingular CM case of the model $C_t$.
24:
-1.800714521389232519501185405743840291737296186591797905532716386162363170058159818083129625765602197
comment: The rational CM parameter has order discriminant $-27$; its modified period is negative. At $t=t_Q([9,3,1])$, $C_t$ has CM order discriminant $\Delta=-27$. The parameter is rational. [2], Section 1. This case also belongs to [1], Table 2; its modified-period identity is proved in [3], Theorem 1.5.
$18-6\sqrt{3}$:
-1.096951909366771601952010997733538752297149349050759564931438930169293673386852406424059633150610001
comment: At $t=t_Q([9,6,2])$, $C_t$ has CM order discriminant $\Delta=-36$. The parameter is real quadratic. [2], Table 1, row 14.
$\frac{-27}{2}+\frac{27}{2}\sqrt{5}$:
-1.513687186239743341872745049688579673087463674158772386230870101576999585487544908803560048817077913
comment: At $t=t_Q([6,3,1])$, $C_t$ has CM order discriminant $\Delta=-15$. The parameter is real quadratic. [2], Table 1, row 17.
$-4320+1944\sqrt{5}$:
-1.932641908064528725581449762187575135462765735637631814047882617171322950109503337595673070721830244
comment: At $t=t_Q([21,3,1])$, $C_t$ has CM order discriminant $\Delta=-75$. The parameter is real quadratic. [2], Table 1, row 19.
$-864+216\sqrt{17}$:
-1.914640409126734184230432088345860797426992831413936376282089809050506578263090011064672112896029418
comment: At $t=t_Q([15,3,1])$, $C_t$ has CM order discriminant $\Delta=-51$. The parameter is real quadratic. [2], Table 1, row 22.
$-163296+35640\sqrt{21}$:
-1.938187616612528602698936496141481210976320713758078558538411926519719123332956344695326978350758086
comment: At $t=t_Q([39,3,1])$, $C_t$ has CM order discriminant $\Delta=-147$. The parameter is real quadratic. [2], Table 1, row 23.
$4+4\sqrt{33}$:
-1.936704632266976978860771242092303076541712282719866196414683070569061135700968614279565508112022658
comment: At $t=t_Q([27,3,1])$, $C_t$ has CM order discriminant $\Delta=-99$. The parameter is real quadratic. [2], Table 1, row 24.
$-16740+2916\sqrt{33}$:
-1.276728224888631527984651430908906310784113775297782331334350359993656517307210947957583367819621887
comment: At $t=t_Q([5,3,1])$, $C_t$ has CM order discriminant $\Delta=-11$. The parameter is real quadratic. [2], Table 1, row 25.
$-55296+8640\sqrt{41}$:
-1.937829950655483536226235978084597768154442686116267964164355182160257613499460155880071043747073648
comment: At $t=t_Q([33,3,1])$, $C_t$ has CM order discriminant $\Delta=-123$. The parameter is real quadratic. [2], Table 1, row 26.
$-13500000+1431000\sqrt{89}$:
-1.938392354885384622642192274639343697785362106957646937023009099408714775556172476069611853799770514
comment: At $t=t_Q([69,3,1])$, $C_t$ has CM order discriminant $\Delta=-267$. The parameter is real quadratic. [2], Table 1, row 27.
1:
-0.5193641873270241569647102574424284335200657978992923046704167352229740920558739013641204423045323859
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
2:
-0.6659452330011374294620998332512325779100619973957945332056222034002327364575949366121530864685656856
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
3:
-0.7721669717878543808248902763318874556126352039922554273362075518130980634213634499331926251917925780
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
4:
-0.8589680019096274110696748111353803361349807092769545576100663765799520751445956665297537994433322078
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
5:
-0.9339770298629909558545952095616790822179847914785896122069195143295305066788226914907841406212626218
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
6:
-1.000953960913923297238908489538500654193897774239851167753975926034530584671097881498999161853718133
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
7:
-1.062068525635516103246392433923184918433615164795440782687396773892994469298188625599175370409783318
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
8:
-1.118704010267936326071994670890374947212412518369189649831947291672637864808724662001107745768467516
comment: Here $k=2$; the modified-period identity with the conductor-19 elliptic curve is proved in [1], Theorem 2.
9:
-1.171806080066140698461301764503744621233423649063652163500911632027918698646439611400329499921813319
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
10:
-1.222056076190743014894964513420593489580546629502493509516741852711565851826751380712220196400984511
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
11:
-1.269965850198057046111197801525241671938940551801142370878867284640552880139832254544409207459702261
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
12:
-1.315933638694070583545848111991270813816601016427280940976081946603490599755515352220404656618612638
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
13:
-1.360279013931852178716074929166985305086579850765493607085781261664472709559267936524366377566675750
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
14:
-1.403265912161444747888196834996851630105587861804779112759558131241669397052202262705739379930080225
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
15:
-1.445118576991690736389685290881880368296892012257995743014961190129388036126211774491504656875479914
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
16:
-1.486033201040458540978468982696718476577870514739646595999458943058731522599877920947928641489926934
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
17:
-1.526186995006989608318395957641507954785463531959131368545035756512494876164078891499049368897706496
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
18:
-1.565745882442955912460800886653019515414602944684463298510289918036365882210331472962521059948339916
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
19:
-1.604871810605948573080603467019987778536338371675434848102950186427486248354548040844653490259490646
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
20:
-1.643730734819702647395371100901982723072627262534938737749510696391361765165047225398387090580479679
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
21:
-1.682502771961712079308896920741846784996874935343267249773453804833732122944094644302712194928027596
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
22:
-1.721397195042573021138178567508465785813208410024946691870031370454041533691605440372475159983816273
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
23:
-1.760677989821890044085571536934937423536124169610457730300912626768451883441648983231422664917215696
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
25:
-1.842103337729136034054286842684580928440125777206798648110888397545628521284667045336481235477139167
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
26:
-1.886070160965905509818387913408177671803341735979082657715441736382115015810787040724351998133341148
comment: An integer parameter of the modified-period arithmetic family in [1], Table 2; the listed $L$-value relation is a numerical proposal, not a proof supplied by this table.
$t_Q([63,3,1])$:
-1.938388688138841055896513461606099186528450986114813595077430218923318397115311961210486341695215492
comment: At $t=t_Q([63,3,1])$, $C_t$ has CM order discriminant $\Delta=-243$. The parameter is real cubic; its other two algebraic conjugates are nonreal. [4], entry ν-C-3-1.
$t_Q([27,24,8])$:
-0.3992711912602014405588042891023534110849460639343881867573198262287790027644195363237575228628554272
comment: At $t=t_Q([27,24,8])$, $C_t$ has CM order discriminant $\Delta=-288$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 52.
$t_Q([27,12,4])$:
-1.625710907153483628775548868204340613335164480005277002196154288014574571729411068721369231437978421
comment: At $t=t_Q([27,12,4])$, $C_t$ has CM order discriminant $\Delta=-288$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 52.
$t_Q([15,12,4])$:
-0.6945825760909440388184795823451764657732161973348647052320759190636915493268695298107736097407016444
comment: At $t=t_Q([15,12,4])$, $C_t$ has CM order discriminant $\Delta=-96$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 53.
$t_Q([21,12,4])$:
-1.348695623404157139536267284798251548044933086077328211651796836844670496337008596977900547029713895
comment: At $t=t_Q([21,12,4])$, $C_t$ has CM order discriminant $\Delta=-192$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 54.
$t_Q([8,8,3])$:
0.3624762913706421220981208095913566862297780276335447056527314560456643150910893825585959113732816759
comment: At $t=t_Q([8,8,3])$, $C_t$ has CM order discriminant $\Delta=-32$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 55.
$t_Q([18,18,7])$:
0.4504538177400977041054262975970502698957692159475982760226704102977743541870185685323623387939151670
comment: At $t=t_Q([18,18,7])$, $C_t$ has CM order discriminant $\Delta=-180$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 57.
$t_Q([27,6,2])$:
-1.903019804545363000745349307309226728910364246904382215726579349483917463342817353181793190634294843
comment: At $t=t_Q([27,6,2])$, $C_t$ has CM order discriminant $\Delta=-180$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 57.
$t_Q([7,6,2])$:
-0.5070736509698308743201140800421117011857372313596145794895069450471123345921855947074727317915365108
comment: At $t=t_Q([7,6,2])$, $C_t$ has CM order discriminant $\Delta=-20$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 58.
$t_Q([15,6,2])$:
-1.703960498856145584623706103638857466776788267714992553617113516886088898567618060450499532943193731
comment: At $t=t_Q([15,6,2])$, $C_t$ has CM order discriminant $\Delta=-84$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 60.
$t_Q([21,6,2])$:
-1.853737266589736050643537188144199338959054597718614028099618407042083119808966924796873101830088095
comment: At $t=t_Q([21,6,2])$, $C_t$ has CM order discriminant $\Delta=-132$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 61.
$t_Q([21,15,5])$:
-0.9593239607041951003975032784115299698415233041581993901682705290715418763133423150918293945393523399
comment: At $t=t_Q([21,15,5])$, $C_t$ has CM order discriminant $\Delta=-195$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 63.
$t_Q([51,3,1])$:
-1.938361121230433534717206069675417178490492478608452053209438861331592437293857788091480065646935218
comment: At $t=t_Q([51,3,1])$, $C_t$ has CM order discriminant $\Delta=-195$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 63.
$t_Q([33,6,2])$:
-1.922121001435714581400827849008090885484173454492122869108220467128721877643159937978327863374356309
comment: At $t=t_Q([33,6,2])$, $C_t$ has CM order discriminant $\Delta=-228$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 66.
$t_Q([27,21,7])$:
-0.7651872760448010359199974975130528130516042917677857450050778079811962408085433125500870476781244270
comment: At $t=t_Q([27,21,7])$, $C_t$ has CM order discriminant $\Delta=-315$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 67.
$t_Q([27,15,5])$:
-1.387257741025163414280326677517788506332799652192827968429997577273288084797679081186858563512017914
comment: At $t=t_Q([27,15,5])$, $C_t$ has CM order discriminant $\Delta=-315$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 67.
$t_Q([81,3,1])$:
-1.938394859486497012284402643950717196959804059408261965656378132135491679630482871461105212716325886
comment: At $t=t_Q([81,3,1])$, $C_t$ has CM order discriminant $\Delta=-315$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 67.
$t_Q([11,3,1])$:
-1.865804620968674501398696149432155792803456157944309350673748957992825132438912705353916661770396131
comment: At $t=t_Q([11,3,1])$, $C_t$ has CM order discriminant $\Delta=-35$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 68.
$t_Q([33,15,5])$:
-1.606481673998718104872982641226405094883462901903445053438686893047249424716698353803679586156951312
comment: At $t=t_Q([33,15,5])$, $C_t$ has CM order discriminant $\Delta=-435$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 69.
$t_Q([111,3,1])$:
-1.938395646254826736696139449676580735413793070908346637134057865617898716496924077810583979960982661
comment: At $t=t_Q([111,3,1])$, $C_t$ has CM order discriminant $\Delta=-435$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 69.
$t_Q([51,6,2])$:
-1.936130790560922082645539054689840888704175193467228152540852134599271334636731838879013352124063991
comment: At $t=t_Q([51,6,2])$, $C_t$ has CM order discriminant $\Delta=-372$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 70.
$t_Q([39,15,5])$:
-1.728543943036135943314644660403291396869788950715718011132197817451920293134198415476913969440538971
comment: At $t=t_Q([39,15,5])$, $C_t$ has CM order discriminant $\Delta=-555$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 71.
$t_Q([141,3,1])$:
-1.938395680856941827816725370036779748413234082861527852532436538791039374846572345320778968610523300
comment: At $t=t_Q([141,3,1])$, $C_t$ has CM order discriminant $\Delta=-555$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 71.
$t_Q([51,15,5])$:
-1.845380351028259542380555159111762541508968003047292073027004070032178478170758753844928052891341789
comment: At $t=t_Q([51,15,5])$, $C_t$ has CM order discriminant $\Delta=-795$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 72.
$t_Q([201,3,1])$:
-1.938395683293945393971844263604139468163200394326135764981217764802666243483074934505955187676147603
comment: At $t=t_Q([201,3,1])$, $C_t$ has CM order discriminant $\Delta=-795$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 72.
$t_Q([93,6,2])$:
-1.938329742130531212843923519486686811499450545115548113790857627819771560698887940881906847890248746
comment: At $t=t_Q([93,6,2])$, $C_t$ has CM order discriminant $\Delta=-708$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 73.
$t_Q([33,21,7])$:
-1.180634857703627126187586399568743564925010268164733142573912433121146816149451140626744766870233194
comment: At $t=t_Q([33,21,7])$, $C_t$ has CM order discriminant $\Delta=-483$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 74.
$t_Q([123,3,1])$:
-1.938395671265168274722011529198950762627384013325373160016410558722279288587485757245129193189537343
comment: At $t=t_Q([123,3,1])$, $C_t$ has CM order discriminant $\Delta=-483$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 74.
$t_Q([39,33,11])$:
-0.5437872510055009646909078986597439717364437945941266194799208780612905330083469944044245414510603207
comment: At $t=t_Q([39,33,11])$, $C_t$ has CM order discriminant $\Delta=-627$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 75.
$t_Q([159,3,1])$:
-1.938395682763875262461780519130645486496886322035447297478101678463811493206009148960563456469731887
comment: At $t=t_Q([159,3,1])$, $C_t$ has CM order discriminant $\Delta=-627$. Its parameter field $\mathbb Q(t)$ is totally real and biquadratic. [2], Table 2, row 75.
$t_Q([4,3,1])$:
-0.8515426369428330004681356443980549239297103811795521825819789931226541938348907468424827090123232767
comment: At $t=t_Q([4,3,1])$, $C_t$ has CM order discriminant $\Delta=-7$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [4], entry ν-N4-1-2.
$t_Q([7,3,1])$:
-1.654863342956889132040496342339957612762091953805086393694806907913471563838841270575395130639645638
comment: At $t=t_Q([7,3,1])$, $C_t$ has CM order discriminant $\Delta=-19$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [4], entry ν-N4-2-2.
$t_Q([13,3,1])$:
-1.897793184257102133753022085942199731633016093723030914822037691201861523344574541816264143826240957
comment: At $t=t_Q([13,3,1])$, $C_t$ has CM order discriminant $\Delta=-43$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [4], entry ν-N4-4-2.
$t_Q([18,3,1])$:
-1.927052409145326726005556409003008402522219004184252742899738451863783411910812298190356613177306856
comment: At $t=t_Q([18,3,1])$, $C_t$ has CM order discriminant $\Delta=-63$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [4], entry ν-N4-5-2.
$t_Q([19,3,1])$:
-1.929402971670142879681496696072588912131633536798894527836427119393456078123000039266305050934850530
comment: At $t=t_Q([19,3,1])$, $C_t$ has CM order discriminant $\Delta=-67$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [4], entry ν-N4-6-2.
$t_Q([43,3,1])$:
-1.938284329880916070006312131236481050425598153731649183661220839865906322734336785482923919160980446
comment: At $t=t_Q([43,3,1])$, $C_t$ has CM order discriminant $\Delta=-163$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [4], entry ν-N4-7-2.
$t_Q([45,3,1])$:
-1.938313343377941097956191950763524170852065535244738374354966721280657499569639293323072939439248640
comment: At $t=t_Q([45,3,1])$, $C_t$ has CM order discriminant $\Delta=-171$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [4], entry ν-N4-8-2.
$t_Q([99,3,1])$:
-1.938395562170711542341744336906820187224232436419644607734282430081709637424120043667560169091387848
comment: At $t=t_Q([99,3,1])$, $C_t$ has CM order discriminant $\Delta=-387$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [4], entry ν-N4-9-2.
$t_Q([153,3,1])$:
-1.938395682416153997463905540908056484347019397121642757271699177706499536939519542896588074563575058
comment: At $t=t_Q([153,3,1])$, $C_t$ has CM order discriminant $\Delta=-603$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [4], entry ν-N4-13-2.
$t_Q([369,3,1])$:
-1.938395683316702307598042761078857879400121366067450437745889732524473721590138299404457190925778472
comment: At $t=t_Q([369,3,1])$, $C_t$ has CM order discriminant $\Delta=-1467$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [4], entry ν-N4-14-2.
$t_Q([12,3,1])$:
-1.884449972655456251245432868169043127895146431505509893983227572782413863498404371814756038727393577
comment: At $t=t_Q([12,3,1])$, $C_t$ has CM order discriminant $\Delta=-39$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [4], entry ν-N4-18-2.
$t_Q([57,3,1])$:
-1.938380451481045591824949030100010936037071754519483194909528471611237268251078580843018543381178587
comment: At $t=t_Q([57,3,1])$, $C_t$ has CM order discriminant $\Delta=-219$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [4], entry ν-N4-19-2.
$t_Q([75,3,1])$:
-1.938394049846275587870555171339931141857725393892140736081123092109003647986923800448726972740795458
comment: At $t=t_Q([75,3,1])$, $C_t$ has CM order discriminant $\Delta=-291$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [4], entry ν-N4-20-2.
$t_Q([93,3,1])$:
-1.938395458452227325913064401968558824158378408659083802680361649904200800367262888829508331375884283
comment: At $t=t_Q([93,3,1])$, $C_t$ has CM order discriminant $\Delta=-363$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [4], entry ν-N4-21-2.
$t_Q([129,3,1])$:
-1.938395676304981258588952164282032380357968612408923806091989384203759763530925061777512733114631460
comment: At $t=t_Q([129,3,1])$, $C_t$ has CM order discriminant $\Delta=-507$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [4], entry ν-N4-22-2.
$t_Q([183,3,1])$:
-1.938395683231263962796819311465690766778916243568664395728114853475349614380470638045583278759557069
comment: At $t=t_Q([183,3,1])$, $C_t$ has CM order discriminant $\Delta=-723$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [4], entry ν-N4-23-2.
$t_Q([309,3,1])$:
-1.938395683316681152863225013989459394796066073149661351442935254295319066470881937238466388391114051
comment: At $t=t_Q([309,3,1])$, $C_t$ has CM order discriminant $\Delta=-1227$. Its quartic parameter field $\mathbb Q(t)$ has two real embeddings and a conjugate pair of nonreal embeddings. [4], entry ν-N4-24-2.
Definition
The modified Mahler measure [1] of the Hessian family is the signed period $\widetilde N(t)=I(t)-2J(t)$ for $-1<t<27$, with the root integrals $I(t),J(t)$ defined in (1).
Parameters
$t$
—   Hessian parameter ($-1<t<27$)
Formulas
(1)
$\displaystyle I(t)=\frac1\pi\int_0^{c(t)}F_t(\theta)\,d\theta,$
$\displaystyle J(t)=\frac1\pi\int_{c(t)}^\pi F_t(\theta)\,d\theta,$
where $k=\sqrt[3]{t}\in\mathbb R$, $c(t)=\arccos((k-1)/2)\in[0,\pi]$
and $F_t(\theta)=\log\max\{|y_1|,|y_2|\}$. Here $y_1,y_2$ are the roots of $Q_t(e^{i\theta},y)=0$, with $Q_t(x,y)=y^2+(x^2-kx)y+x$ [1]. The variables $x,y$ are integration variables.
(2)
$N(t)=I(t)+J(t),$
$\widetilde N(t)=I(t)-2J(t)=N(t)-3J(t).$
Here $N(t)$ is the ordinary Mahler measure, and $I(t),J(t)$ are the two root integrals in (1) [2], Section 2.
(3)
For $0<t<27$ and $k=\sqrt[3]{t}$,
$\displaystyle\begin{aligned}\widetilde N(t)={}&-\frac{\Gamma(1/3)^3 k}{4\pi^2}\,{}_3F_2\!\left(\begin{matrix}\frac13,\frac13,\frac13\\\frac23,\frac43\end{matrix};\frac{t}{27}\right)\\&-\frac{\Gamma(2/3)^3 k^2}{8\pi^2}\,{}_3F_2\!\left(\begin{matrix}\frac23,\frac23,\frac23\\\frac43,\frac53\end{matrix};\frac{t}{27}\right).\end{aligned}$
This is the convergent-series form following Theorem 1 of [1], with its first coefficient simplified by the gamma duplication and reflection formulas. The constants $\Gamma(1/3)$ and $\Gamma(2/3)$ are tabulated separately.
(4)
$\displaystyle\begin{aligned}C_t:\quad Y^2={}&X^3-27t^2X^2\\&+216t^3(t-27)X-432t^4(t-27)^2,\end{aligned}\qquad j(C_t)=\frac{t(t+216)^3}{(t-27)^3}. $ The model is nonsingular for $t\ne0,27$. If $t=t_Q([A,B,C])$, then $j(C_t)=j(\tau_Q([A,B,C]))$ and $H_\Delta(j(C_t))=0$, where $H_\Delta$ is the class polynomial of the imaginary quadratic order of discriminant $\Delta$ [2], Lemmas 2.3 and 2.4. The class-polynomial table contains examples for some of these orders, not necessarily every discriminant in the expanded catalogue.
(5)
$\displaystyle \tau_Q([A,B,C])=\frac{-B+i\sqrt{4AC-B^2}}{2A},\qquad t_Q(\tau)=27+\left(\frac{\eta(\tau)}{\eta(3\tau)}\right)^{12}. $ The shorthand $t_Q([A,B,C])$ means $t_Q(\tau_Q([A,B,C]))$. Here $A,B,C\in\mathbb Z$, $A>0$, $\gcd(A,B,C)=1$ and $\Delta=B^2-4AC<0$. The Dedekind eta function is $\eta(\tau)=e^{\pi i\tau/12}\prod_{n\geq1}(1-e^{2\pi i n\tau})$ for $\operatorname{Im}\tau>0$. The level-three representatives follow [2], Sections 2 and 3.
(6)
For $-1<t<0$ and the real cube root $k=\sqrt[3]{t}<0$, $\displaystyle \widetilde N(t)=\operatorname{Re}\left[\log k-\frac{2}{t}\,{}_4F_3\!\left(\begin{matrix}\frac43,\frac53,1,1\\2,2,2\end{matrix};\frac{27}{t}\right)\right].$ This is Theorem 1 of [1] on its negative branch: $4/(1-3\operatorname{sgn}k)=1$. The logarithm is the principal complex logarithm before taking the real part, and ${}_4F_3$ is continued from its series to the negative real axis.
(7)
$\widetilde N(0)=0.$ This is the continuous value of $I(t)-2J(t)$ at the Fermat parameter, also obtained from the positive-side series in (3). It is different from $N(0)$.
Comments
(8)
Natural logarithms are used. The product of the two roots is $e^{i\theta}$, so the larger root has modulus at least one; a tie gives $F_t(\theta)=0$. This agrees with Samart's larger-root branch [1], Lemma 3, and remains defined at the removable singularity of the explicit root formula when $k=1$.
(9)
The ordinary logarithmic Mahler measure $N(t)$ of $P_t=x^3+y^3+1-kxy$ equals that of $Q_t=y^2+(x^2-kx)y+x$. With $I(t)=\pi^{-1}\int_0^cF_t(\theta)\,d\theta$ and $J(t)=\pi^{-1}\int_c^\pi F_t(\theta)\,d\theta$, the ordinary and modified periods are the different combinations in (2). The sign of the modified period is retained.
(10)
The parameter is $t=k^3$, with the real cube root fixing $k$. Samart writes $\tilde n(k)$ [1]; Samart and Tao use $\tilde{\mathrm n}(t)$ and the piecewise symbol $\nu(t)$ [2]. This table holds the modified period on its open real domain.
(11)
The CM arguments are the 57 real parameters in $(-1,27)$ from the degree-at-most-four classification in [2] and [4], including two negative arguments. Real embeddings of non-totally-real fields are included. Complex multiplication refers to $C_t$ in (4); the degree is that of $t$, not its cube root.
(12)
In $t_Q([A,B,C])$, the triple specifies the CM point $\tau_Q([A,B,C])$ in (5), while $t_Q$ is the level-three modular function giving the Hessian argument. The order discriminant $\Delta=B^2-4AC$ belongs to an imaginary quadratic order; it is not the discriminant of the parameter field $\mathbb Q(t)$. Neither $\Delta$ nor the field degree alone identifies the argument or its embedding.
(13)
The integers $1\leq t\leq26$ form the arithmetic reference family in [1], Table 2, rather than an interpolation grid. Their proposed $L$-value identities are not all proved. The case $t=8$ is proved in [1], Theorem 2, and $t=24$ in [3], Theorem 1.5.
(14)
For negative arguments, the ${}_3F_2$ prefactor printed after Theorem 1 of [1] gives one quarter of the defining period in the checked examples, whereas the theorem's general ${}_4F_3$ expression agrees with the defining integrals. The negative branch is therefore stated directly as (6); (3) remains restricted to $0<t<27$. This is an observed source inconsistency, not an author-confirmed erratum.
Programs
(P1)
Sage
from generate import Table

table = Table()
print(table.value({'t': '24'}, 100)['number'])
References
[1]
D. Samart, Mahler measure of a nonreciprocal family of elliptic curves, corrected version 3 (2023). (arXiv)
[2]
D. Samart and Z. Tao, Determinants of Mahler measures and special values of L-functions, version 3 (2026). (arXiv)
[3]
Z. Tao and X. Guo, CM points, class numbers, and the Mahler measures of x^3+y^3+1-kxy, version 2 (2024). (arXiv)
Links
Similar tables
Mahler measures of the Hessian family —   holds the ordinary period $I(t)+J(t)$ for the same polynomial and parameter normalization, whereas this table holds $I(t)-2J(t)$
Hilbert_class_polynomials —   holds class polynomials that identify CM orders; its present coverage need not include every discriminant in this expanded parameter catalogue
Values_of_the_Gamma_function_at_rational_numbers —   holds the gamma constants in the convergent hypergeometric evaluation of the modified period
Quadratic CM determinants of Hessian Mahler measures —   forms $D(t)=a\nu(t)^2+b\nu(t^\sigma)^2$ at oriented quadratic CM arguments with fixed coefficients $a,b$; the conjugate inputs in $(-1,27)$ use the signed modified period $\nu(u)=\widetilde N(u)$ from this table, while exterior inputs use the ordinary measure
Data properties
Entries are of type: real number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds 83 distinct arguments: all 57 real CM parameters of degree at most four lying in $(-1,27)$ from [2] and [4], the integers $1\leq t\leq26$ from [1], Table 2, and the intrinsic value at $t=0$. The overlap at $t=24$ is counted once; the open interval and its remaining real arguments are not exhausted)
How they were obtained:

The attached generator evaluates the quadratic Jensen integral by adaptive tanh-sinh quadrature, split at the torus crossing and at geometrically spaced points around the narrow unit-modulus transition near $t=27$. A stable real expression avoids subtraction loss in the root-modulus integrand. For 100-digit output it uses nominal precisions of 140 and 190 digits with 40 extra integration digits (180 and 230 effective working digits), retaining 135 and 185 intermediate digits. The estimates and precision comparisons are heuristic, not certified error bounds.

more

All 83 entries were recomputed; nonzero values agree at the two nominal precisions to at least 134.33 relative decimal digits. The table retains 100 significant digits. The value at $t=0$ is the exact integer zero.

An independent hypergeometric implementation at nominal precisions of 125 and 145 digits uses a logarithmic series near $t=27$ with 20 extra working digits. Negative arguments use additional working digits recorded in the per-entry evidence. The independent computation agrees with every nonzero modified value to over 110 relative digits; it also verifies the exact zero. Negative arguments use Theorem 1 of [1], not the inconsistent printed negative prefactor discussed in the comments.

The CM arguments are reconstructed from integer minimal polynomials and exact rational isolating intervals. Exact class-polynomial relations and independent eta-product enclosures identify their level-three CM forms and embeddings. All 66 order discriminants were checked with Sage/Arb, PARI and direct ball products; 49 overlap the present class-polynomial table. These argument checks do not certify the numerical period integrals.

The source hashes, complete checks and preservation tests are in verification.json. Earlier attached numerical reports describe their original selections; argument-selection.md distinguishes them from the expanded revision.



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