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{"Title": "Mahler measures of the Hessian family", "Definition": "The ordinary logarithmic Mahler measure CITE{Wiki} of the Hessian cubic $P_t=x^3+y^3+1-\\sqrt[3]{t}xy$ CITE{S} is $N(t)=\\frac{1}{(2\\pi)^2}\\int_0^{2\\pi}\\int_0^{2\\pi}\\log|P_t(e^{i\\theta},e^{i\\phi})|\\,d\\theta\\,d\\phi$, using the real cube root for $t\\in\\mathbb R$.", "Parameters": {"t": {"type": "R", "display": "$t$", "title": "Hessian parameter", "constraints": "$t\\in\\mathbb R$"}}, "Comments": {"comment-1": "Natural logarithms and normalized Haar measure are used. Zeros of $P_t$ on the torus are interpreted through the convergent logarithmic integral. In particular, for $-1<t<27$, $N(t)$ is the ordinary measure, which differs from the modified period introduced in CITE{S}.", "comment-2": "The parameter is the cube of the coefficient $k$. Samart writes $n(k)$ CITE{S}, whereas Samart and Tao write $\\mathrm n(t)$ CITE{ST}. The real cube root fixes the convention for negative $t$.", "comment-3": "The arithmetic arguments are the three rational CM parameters of the model $C_t$ in CITE{formula-cm}, $-216,24,54$ and both real embeddings of the fifteen quadratic parameters in CITE{ST}, Table 1, rows 13 through 27. The degree bound applies to $t$, not $k$. The CM discriminants and exact radical identities are recorded in the attached generator and argument-selection note.", "comment-4": "At $t=0$, the torus covering $(x,y)\\mapsto(x^3,y^3)$ gives $N(0)=\\mu_3$, the three-step value in HREF{Mahler_measures_of_(short_random_walks)#3}. At $t=27$, the factorization of the cubic into three linear factors gives $N(27)=3\\mu_3$."}, "Formulas": {"formula-jensen": "$N(t)=m\\bigl(y^2+(x^2-kx)y+x\\bigr)=\\frac1\\pi\\int_0^\\pi\\log\\max\\{|y_1(e^{i\\theta})|,|y_2(e^{i\\theta})|\\}\\,d\\theta,$ Here $k=\\sqrt[3]{t}$ is real and $y_1(x),y_2(x)$ are the roots in $y$ of $y^2+(x^2-kx)y+x$. Their product is $x$, so on the unit circle the larger root has modulus at least one CITE{S}, Lemma 3.", "formula-controls": "$N(0)=\\mu_3=\\frac{3\\sqrt3}{4\\pi}L(2,\\chi_{-3}),\\qquad N(27)=3\\mu_3,$ Here $\\mu_3$ is HREF{Mahler_measures_of_(short_random_walks)#3}[the three-step Mahler measure] and $\\chi_{-3}(n)$ is $0,1,-1$ for $n\\equiv0,1,2\\pmod3$, respectively. The value HREF{Values_of_Dirichlet_L-functions_at_positive_integers#3,2,2}[$L(2,\\chi_{-3})$] uses Conrey label $(3,2)$.", "formula-cm": "$j(C_t)=\\frac{t(t+216)^3}{(t-27)^3},\\qquad C_t:\\ Y^2=X^3-27t^2X^2+216t^3(t-27)X-432t^4(t-27)^2,$ This model is nonsingular for $t\\ne0,27$ CITE{ST}. A selected CM parameter satisfies $H_D(j(C_t))=0$ for its imaginary quadratic order discriminant $D$; HREF{Hilbert_class_polynomials} holds these polynomials."}, "Programs": {"program-sage": {"language": "Sage", "code": "from generate import HessianMahler\ng = HessianMahler()\ng.value({'t': '24'}, 100)", "comment": "With the attached generate.py in the working directory, this computes the ordinary measure at the rational CM parameter 24. The default enumeration reproduces the complete 36-entry selection."}}, "Similar tables": [{"table": "HREF{Mahler_measures_of_(short_random_walks)}", "relation": "holds $\\mu_3=N(0)$, also giving the singular-fibre value $N(27)=3\\mu_3$"}, {"table": "HREF{Mahler_measures_of_x_x-1_y_y-1_k}", "relation": "holds logarithmic Mahler measures of the reciprocal Laurent family $x+x^{-1}+y+y^{-1}+k$"}, {"table": "HREF{Hilbert_class_polynomials}", "relation": "holds the polynomials $H_D$ that characterize the CM $j$-invariants of the selected Hessian curves"}, {"table": "HREF{Values_of_Dirichlet_L-functions_at_positive_integers}", "relation": "holds $L(2,\\chi_{-3})$, the Dirichlet value in the evaluations at $t=0$ and $t=27$"}], "Links": {"Wiki": {"title": "Mahler measure", "url": "https://en.wikipedia.org/wiki/Mahler_measure"}}, "References": {"S": {"bib": "D. Samart, Mahler measure of a nonreciprocal family of elliptic curves, corrected version 3 (2023).", "arxiv": "2301.05390v3"}, "ST": {"bib": "D. Samart and Z. Tao, Determinants of Mahler measures and special values of L-functions, version 3 (2026).", "arxiv": "2409.14714v3"}}, "Keywords": ["Hesse", "Hessian", "Mahler measure", "x^3+y^3+1-kxy", "CM", "Samart", "Samart-Tao"], "Tags": ["number theory", "special values", "elliptic curves", "integral"], "Data properties": {"type": "R", "rigour": "heuristic (agreement-checked)", "digits": "100", "complete": "no", "complete-note": "it holds the intrinsic controls $t=-1,0,27$, the three rational CM parameters listed in CITE{ST}, and both embeddings of every quadratic parameter in CITE{ST}, Table 1, rows 13 through 27, for 36 entries; the degree-two catalogue is a noncanonical computational boundary chosen to cover its arithmetic orbits before moving to quartic fields, and no continuous reference mesh is used", "rigour details": "The generator uses adaptive tanh-sinh quadrature of the Jensen integral CITE{formula-jensen}. It splits at the torus crossing $\\theta=\\arccos((k-1)/2)$ when $-1<t<27$, and bisects intervals whose numerical convergence estimates are insufficient. All 36 values were computed at 140 and 190 working digits, with at least 136.04 digits of relative agreement. The table retains 100 significant digits. These are agreement-checked approximations: the quadrature estimates are not certified error bounds.\n\nAn independent implementation applies Jensen's formula directly to $x^3+y^3+1-kxy$, solving the cubic in $y$ and integrating the sum of the positive logarithms of its root moduli at 125 working digits. It agrees with all 36 results to at least 119.80 relative digits. A separate hypergeometric computation checks 24 arguments outside the open torus-zero interval; at $t=27$, the factorization gives the independent control $3N(0)$. The cubic integrator first reproduces the nonzero control $N(0)=3\\sqrt3 L(2,\\chi_{-3})/(4\\pi)$, evaluated using Hurwitz zeta functions.\n\nThe 33 CM arguments are constructed exactly as real algebraic numbers. Their $j$-invariants satisfy the corresponding HREF{Hilbert_class_polynomials}[Hilbert class polynomials] in exact arithmetic; the polynomials were also recomputed with Sage. The model's $j$-invariant formula was checked symbolically. The attached checks, numerical outputs and argument-selection note record the conventions, precision comparisons and catalogue coverage."}, "Display properties": {"number-header": "$N(t)$"}, "Numbers": {"-1": {"number": "0.4549624460247022157899646290452669704837710430448280179561721677447580407691966814707910045615975207", "comment": "The negative real boundary of the torus-zero region, with $k=-1$."}, "0": {"number": "0.3230659472194505140936365107238063940722418407805870161308684703610151128072698420837876090893713921", "comment": "The Fermat specialization gives $N(0)=\\mu_3$, the three-step value in HREF{Mahler_measures_of_(short_random_walks)#3}."}, "27": {"number": "0.9691978416583515422809095321714191822167255223417610483926054110830453384218095262513628272681141762", "comment": "The cubic factors into three linear factors, each of Mahler measure $\\mu_3$, so $N(27)=3\\mu_3$."}, "-216": "1.800714521389232519501185405743840291737296186591797905532716386162363170058159818083129625765602197", "24": "0.9017689717714497038711883043277513920447874379412718951342713018471165138878658154816614965952738831", "54": "1.285783612394876596075276669356046539933373792348389251008786266766574524229329991694179167604592026", "108+54*sqrt(2)": {"number": "1.727655340123065766541154958654423253125575351078219975610305918610812048989139033331494265061730081", "param-latex": "$108+54\\sqrt{2}$"}, "108-54*sqrt(2)": {"number": "1.059905935890393452450890400849350487614758432820326968317339530385018099053017691335844399986334278", "param-latex": "$108-54\\sqrt{2}$"}, "18+6*sqrt(3)": {"number": "1.003029834811605129839867869536644140453242834722302446325158096488593803194804733989674878283400100", "param-latex": "$18+6\\sqrt{3}$"}, "18-6*sqrt(3)": {"number": "0.5920965469947062494295412092683636467226951844411848086153302840833862418377243337219524125987214725", "param-latex": "$18-6\\sqrt{3}$"}, "270+162*sqrt(3)": {"number": "2.099981744178376731791878867270182892750392183773062011256597026657887476581657140413734511434010101", "param-latex": "$270+162\\sqrt{3}$"}, "270-162*sqrt(3)": {"number": "0.9091077602564386577277247413397495286093363203938453277188772628078939330027570615552901234161901998", "param-latex": "$270-162\\sqrt{3}$"}, "729+405*sqrt(3)": {"number": "2.420516385163434449629413366858139565791557279776831965552730062644549948187330118305482191174311361", "param-latex": "$729+405\\sqrt{3}$"}, "729-405*sqrt(3)": {"number": "0.9836819331422391110329967256481395117829931825084961780791013249893763236866702615284268795348466448", "param-latex": "$729-405\\sqrt{3}$"}, "-27/2-27/2*sqrt(5)": {"number": "1.298605737392881591792249284500453734684030523082832218706201803316600428159594233358650806356989239", "param-latex": "$\\frac{-27}{2}-\\frac{27}{2}\\sqrt{5}$"}, "-27/2+27/2*sqrt(5)": {"number": "0.7680434591650134888193730218685892439272060749216783350865008980102679556463861436772083721875979691", "param-latex": "$\\frac{-27}{2}+\\frac{27}{2}\\sqrt{5}$"}, "3375/2+1485/2*sqrt(5)": {"number": "2.704752199209194058624746451594970438569777621703634521174737755763400434983163804439756233944022815", "param-latex": "$\\frac{3375}{2}+\\frac{1485}{2}\\sqrt{5}$"}, "3375/2-1485/2*sqrt(5)": {"number": "0.9759835641225889666715056367182648270788807964689219674191993559260016921676682201912869426668562285", "param-latex": "$\\frac{3375}{2}-\\frac{1485}{2}\\sqrt{5}$"}, "-4320-1944*sqrt(5)": {"number": "3.022653334601145585262377400398605184457541948537194128053826472344298815730259755066552833912772892", "param-latex": "$-4320-1944\\sqrt{5}$"}, "-4320+1944*sqrt(5)": {"number": "0.9663270874013787041406197662486857338628364712017570767216309787834215036807874658818125360598719301", "param-latex": "$-4320+1944\\sqrt{5}$"}, "32+2*sqrt(6)": {"number": "1.130631303161317244951209819006588591957195612395167473278322660001297126566378926469503558130560769", "param-latex": "$32+2\\sqrt{6}$"}, "32-2*sqrt(6)": {"number": "0.9725915934916164598735756838801957322347906660861116463548964739468381202515279599165171961333108118", "param-latex": "$32-2\\sqrt{6}$"}, "3672+1458*sqrt(6)": {"number": "2.962336905206945724844014805642105291978386650950761319767718364651972668738726815067680701015745399", "param-latex": "$3672+1458\\sqrt{6}$"}, "3672-1458*sqrt(6)": {"number": "1.515626182706226909737536716382730980576381145937482192642543332541757529542645955492204418890684715", "param-latex": "$3672-1458\\sqrt{6}$"}, "-864-216*sqrt(17)": {"number": "2.491132013779120844135437254110611891211852906172703926477410677344981893069857893797590982354393743", "param-latex": "$-864-216\\sqrt{17}$"}, "-864+216*sqrt(17)": {"number": "0.9573868395346952929596114534383981915360180479862203843028483173350888318620065904514728450276440335", "param-latex": "$-864+216\\sqrt{17}$"}, "-163296-35640*sqrt(21)": {"number": "4.232189331912333939917662110313971819477043143599539860830536987919254536131624970969316424092532517", "param-latex": "$-163296-35640\\sqrt{21}$"}, "-163296+35640*sqrt(21)": {"number": "0.9690938348865934061032467741226386558059164917456072943424975934076455079376955618820055877776451835", "param-latex": "$-163296+35640\\sqrt{21}$"}, "4-4*sqrt(33)": {"number": "1.060313853042720113357474149455609607271032448715052961574037589506501775789646915487615814676119601", "param-latex": "$4-4\\sqrt{33}$"}, "4+4*sqrt(33)": {"number": "0.9683531251520881988303042692441578703441717364098282766471597927345970196613304981384267750852166606", "param-latex": "$4+4\\sqrt{33}$"}, "-16740-2916*sqrt(33)": {"number": "3.473071818869579295240188145912393311569990624146763296335661262207025114006047979393497341840326488", "param-latex": "$-16740-2916\\sqrt{33}$"}, "-16740+2916*sqrt(33)": {"number": "0.6650239619828272423587108944358792115552304440499644641099408088759290276016741318065889380791871040", "param-latex": "$-16740+2916\\sqrt{33}$"}, "-55296-8640*sqrt(41)": {"number": "3.871300440182100051356811451853387873673770608739135498460201822237140445768628320239310401861257802", "param-latex": "$-55296-8640\\sqrt{41}$"}, "-55296+8640*sqrt(41)": {"number": "0.9689151101674892016309712836393120891491166353520058919461376976098160939137379280693304222179265571", "param-latex": "$-55296+8640\\sqrt{41}$"}, "-13500000-1431000*sqrt(89)": {"number": "5.703782882063013817942765984840551372015462014636724568739699272076956711941714749660016641013224744", "param-latex": "$-13500000-1431000\\sqrt{89}$"}, "-13500000+1431000*sqrt(89)": {"number": "0.9691961774780599931093656328095021079956638816652182588008551473090827020184711084142924091287484464", "param-latex": "$-13500000+1431000\\sqrt{89}$"}}}