{"Title": "Quadratic CM determinants of Hessian Mahler measures", "Definition": "The quadratic CM Mahler determinant CITE{ST} is $D(t)=a\\nu(t)^2+b\\nu(t^\\sigma)^2$, with the oriented parameter and fixed coefficients in CITE{formula-expanded}.", "Parameters": {"t": {"type": "R", "display": "$t$", "title": "oriented Hessian CM parameter", "constraints": "$t$ is a real quadratic CM parameter of $C_t$, with its embedding and determinant normalization fixed as in CITE{ST}."}}, "References": {"ST": {"bib": "D. Samart and Z. Tao, Determinants of Mahler measures and special values of L-functions, version 3 (2026).", "arxiv": "2409.14714v3"}, "S": {"bib": "D. Samart, Mahler measure of a nonreciprocal family of elliptic curves, corrected version 3 (2023).", "arxiv": "2301.05390v3"}, "TG": {"bib": "Z. Tao and X. Guo, CM points, class numbers, and the Mahler measures of x^3+y^3+1-kxy, version 2 (2024).", "arxiv": "2310.12510v2"}}, "Links": {"Wiki": {"title": "Mahler measure", "url": "https://en.wikipedia.org/wiki/Mahler_measure"}}, "Tags": ["number theory", "special values", "elliptic curves", "integral", "L-function"], "Data properties": {"type": "R", "digits": "100", "rigour": "heuristic (agreement-checked)", "complete": "no", "complete-note": "it holds all fifteen oriented real quadratic representatives with their prescribed determinant coefficients from CITE{ST}, Table 1, rows 13 through 27; the degree-two catalogue is a noncanonical computational boundary chosen to cover its arithmetic orbits before quartic matrices, and no continuous reference mesh is used", "rigour details": "The generator evaluates the two scalar periods analytically and forms the signed determinant. Outside the torus-zero interval it uses the constant-term series, with the equivalent hypergeometric continuation for the selected negative arguments of absolute value at most 27. Inside the interval it uses the corrected convergent hypergeometric identity of CITE{S}, or integrates its derivative expressed through two Gauss hypergeometric functions near the singular fibre. Exact radical arguments are reconstructed at each working precision. All fifteen determinants were evaluated at 140 and 190 working decimal digits, with at least 131.39 relative digits of agreement; 100 significant digits are retained. These are agreement-checked approximations, not certified error bounds.\n\nAn independent implementation solves the defining quadratic on the unit circle and integrates the logarithm of the larger root modulus, split at the torus crossing for interior arguments. It uses 125 working digits and first reproduces the nonzero control HREF{T649#0}[$N(0)$] from a Hurwitz-zeta evaluation of HREF{Values_of_Dirichlet_L-functions_at_positive_integers#3,2,2}[$L(2,\\chi_{-3})$]. All thirty signed scalar inputs and all fifteen determinants agree with the analytic computation to at least 114.41 relative digits. The largest cancellation factor, the sum of the absolute quadratic terms divided by the absolute determinant, is less than 1.087.\n\nBoth embeddings of all fifteen arguments satisfy their exact Hilbert class polynomials, which are independently recomputed and compared with HREF{Hilbert_class_polynomials}. The model $j$-invariant is checked symbolically. The arithmetic identity CITE{formula-row16} is separately evaluated using PARI elliptic-curve L-functions at 140 and 190 working digits; it agrees to more than 181 relative digits. The elliptic curves reproduce the published initial newform coefficients. The attached generator, checks, argument-selection note and numerical evidence record the scope, conventions and measured agreement."}, "Display properties": {"number-header": "$D(t)$"}, "Numbers": {"108+54*sqrt(2)": {"number": "7.092986541447182950242847706092825572773193761643221036343250881638249829759358487071944225821788192", "param-latex": "$108+54\\sqrt{2}$", "comment": "The coefficients are $a=2$ and $b=1$ (CITE{ST}, Table 1, row 13). The period at $t$ is HREF{T649#108+54*sqrt(2)}[$N$] and the period at $t^\\sigma$ is HREF{T649#108-54*sqrt(2)}[$N$]."}, "18+6*sqrt(3)": {"number": "3.215441190507797646935744825321917295105956335366857719546111742743651310674569693262700183971555823", "param-latex": "$18+6\\sqrt{3}$", "comment": "The coefficients are $a=2$ and $b=1$ (CITE{ST}, Table 1, row 14). The period at $t$ is HREF{T649#18+6*sqrt(3)}[$N$] and the period at $t^\\sigma$ is HREF{T650#18-6*sqrt(3)}[$\\widetilde N$]."}, "270+162*sqrt(3)": {"number": "9.646323571523392940807234475965751885317869006100573158638335228230953932023709079788100551914667469", "param-latex": "$270+162\\sqrt{3}$", "comment": "The coefficients are $a=2$ and $b=1$ (CITE{ST}, Table 1, row 15). The period at $t$ is HREF{T649#270+162*sqrt(3)}[$N$] and the period at $t^\\sigma$ is HREF{T649#270-162*sqrt(3)}[$N$]."}, "729+405*sqrt(3)": {"number": "22.46796813778818642931994948020389706049816774683251140180796289188299013168800016813735310496534755", "param-latex": "$729+405\\sqrt{3}$", "comment": "The coefficients are $a=4$ and $b=-1$ (CITE{ST}, Table 1, row 16). The period at $t$ is HREF{T649#729+405*sqrt(3)}[$N$] and the period at $t^\\sigma$ is HREF{T649#729-405*sqrt(3)}[$N$]."}, "-27/2-27/2*sqrt(5)": {"number": "10.72313320373494018150350977439978658466128463871421104376126607645189949646694950294938334208542441", "param-latex": "$\\frac{-27}{2}-\\frac{27}{2}\\sqrt{5}$", "comment": "The coefficients are $a=5$ and $b=1$ (CITE{ST}, Table 1, row 17). The period at $t$ is HREF{T649#-27/2-27/2*sqrt(5)}[$N$] and the period at $t^\\sigma$ is HREF{T650#-27/2+27/2*sqrt(5)}[$\\widetilde N$]."}, "3375/2+1485/2*sqrt(5)": {"number": "37.53096621307229063526228421039925304631449623549973865316443126758164823763432326032284169729898543", "param-latex": "$\\frac{3375}{2}+\\frac{1485}{2}\\sqrt{5}$", "comment": "The coefficients are $a=5$ and $b=1$ (CITE{ST}, Table 1, row 18). The period at $t$ is HREF{T649#3375/2+1485/2*sqrt(5)}[$N$] and the period at $t^\\sigma$ is HREF{T649#3375/2-1485/2*sqrt(5)}[$N$]."}, "-4320-1944*sqrt(5)": {"number": "224.6757247845783219820084691590401519305673376921029709373504752518183988426475211259287440756456580", "param-latex": "$-4320-1944\\sqrt{5}$", "comment": "The coefficients are $a=25$ and $b=-1$ (CITE{ST}, Table 1, row 19). The period at $t$ is HREF{T649#-4320-1944*sqrt(5)}[$N$] and the period at $t^\\sigma$ is HREF{T650#-4320+1944*sqrt(5)}[$\\widetilde N$]."}, "32+2*sqrt(6)": {"number": "3.502588695107078646422882472997385988169702474991076464997298816933667951825228498024783819205000639", "param-latex": "$32+2\\sqrt{6}$", "comment": "The coefficients are $a=2$ and $b=1$ (CITE{ST}, Table 1, row 20). The period at $t$ is HREF{T649#32+2*sqrt(6)}[$N$] and the period at $t^\\sigma$ is HREF{T649#32-2*sqrt(6)}[$N$]."}, "3672+1458*sqrt(6)": {"number": "19.84800260560677899639633401365185393296164735828276663498469329595745172700962815547377497549500362", "param-latex": "$3672+1458\\sqrt{6}$", "comment": "The coefficients are $a=2$ and $b=1$ (CITE{ST}, Table 1, row 21). The period at $t$ is HREF{T649#3672+1458*sqrt(6)}[$N$] and the period at $t^\\sigma$ is HREF{T649#3672-1458*sqrt(6)}[$N$]."}, "-864-216*sqrt(17)": {"number": "109.1634059675396927569206469171382947552509253067376938515398601944019003292932366917923776570625801", "param-latex": "$-864-216\\sqrt{17}$", "comment": "The coefficients are $a=17$ and $b=1$ (CITE{ST}, Table 1, row 22). The period at $t$ is HREF{T649#-864-216*sqrt(17)}[$N$] and the period at $t^\\sigma$ is HREF{T650#-864+216*sqrt(17)}[$\\widetilde N$]."}, "-163296-35640*sqrt(21)": {"number": "873.9033292792856531182314015727567039171374015464944235719340742008187516874623002450580743439875301", "param-latex": "$-163296-35640\\sqrt{21}$", "comment": "The coefficients are $a=49$ and $b=-1$ (CITE{ST}, Table 1, row 23). The period at $t$ is HREF{T649#-163296-35640*sqrt(21)}[$N$] and the period at $t^\\sigma$ is HREF{T650#-163296+35640*sqrt(21)}[$\\widetilde N$]."}, "4-4*sqrt(33)": {"number": "16.11774496914165624216791929014262408143631171882729812301796159647007002786847706274513275220933787", "param-latex": "$4-4\\sqrt{33}$", "comment": "The coefficients are $a=11$ and $b=1$ (CITE{ST}, Table 1, row 24). The period at $t$ is HREF{T649#4-4*sqrt(33)}[$N$] and the period at $t^\\sigma$ is HREF{T650#4+4*sqrt(33)}[$\\widetilde N$]."}, "-16740-2916*sqrt(33)": {"number": "134.3145414095138020180659940845218673453025976568941510251496799705839168989039755228761062684111489", "param-latex": "$-16740-2916\\sqrt{33}$", "comment": "The coefficients are $a=11$ and $b=1$ (CITE{ST}, Table 1, row 25). The period at $t$ is HREF{T649#-16740-2916*sqrt(33)}[$N$] and the period at $t^\\sigma$ is HREF{T650#-16740+2916*sqrt(33)}[$\\widetilde N$]."}, "-55296-8640*sqrt(41)": {"number": "618.2208359419764200914664915370125555404604865786987689205508265792866246770171059802078995552867217", "param-latex": "$-55296-8640\\sqrt{41}$", "comment": "The coefficients are $a=41$ and $b=1$ (CITE{ST}, Table 1, row 26). The period at $t$ is HREF{T649#-55296-8640*sqrt(41)}[$N$] and the period at $t^\\sigma$ is HREF{T650#-55296+8640*sqrt(41)}[$\\widetilde N$]."}, "-13500000-1431000*sqrt(89)": {"number": "2899.206750670118464346726431718974939988615770205808065544853280436864224226453978998417683054618882", "param-latex": "$-13500000-1431000\\sqrt{89}$", "comment": "The coefficients are $a=89$ and $b=1$ (CITE{ST}, Table 1, row 27). The period at $t$ is HREF{T649#-13500000-1431000*sqrt(89)}[$N$] and the period at $t^\\sigma$ is HREF{T650#-13500000+1431000*sqrt(89)}[$\\widetilde N$]."}}, "Comments": {"comment-normalization": "The square roots in the parameter labels are positive. Each entry gives its prescribed integers $a,b$ and source row in CITE{ST}, Table 1. They are fixed normalization data, not independently variable arguments. The determinant retains its sign; no absolute value or primitive rescaling is applied.", "comment-periods": "The ordinary logarithmic Mahler measure CITE{Wiki} HREF{T649}[$N(t)$] and the modified period HREF{T650}[$\\widetilde N(t)$] use natural logarithms and the real cube root $k=\\sqrt[3]{t}$, including for negative $t$. Their defining polynomial is $P_t(x,y)=x^3+y^3+1-kxy$. The determinant uses the modified period only in the open interval $(-1,27)$, according to CITE{formula-period}.", "comment-cm": "Complex multiplication refers to the curve $C_t$ in CITE{formula-cm}. The selected real quadratic arguments are the fifteen oriented representatives in CITE{ST}, Table 1, rows 13 through 27. Their conjugates occur as scalar inputs in the determinant. The field degree is the degree of $t$, not of its real cube root.", "comment-L": "The determinant normalization comes from the arithmetic identities of CITE{ST}, Theorem 5.2. The special case $t=729+405\\sqrt3$ also appears in CITE{TG}, Theorem 1.4, whose scalar convention is $m_3=3N$; its two-dimensional determinant is therefore nine times $D(t)$."}, "Formulas": {"formula-expanded": "$D(t)=\\det\\begin{pmatrix}a\\nu(t)&b\\nu(t^\\sigma)\\\\-\\nu(t^\\sigma)&\\nu(t)\\end{pmatrix}=a\\nu(t)^2+b\\nu(t^\\sigma)^2.$ Here $t^\\sigma$ is the algebraic conjugate of the oriented real quadratic CM argument $t$, $a,b$ are its fixed normalization coefficients from CITE{ST}, Table 1, and $\\nu$ is the period in CITE{formula-period}. This is the normalization of CITE{ST}, equation (5.5).", "formula-period": "$\\displaystyle \\nu(t)=\\begin{cases}\\widetilde N(t),&-1<t<27,\\\\N(t),&t\\leq-1\\ \\text{or}\\ t\\geq27.\\end{cases}$ Here HREF{T649}[$N$] is the ordinary Hessian Mahler measure and HREF{T650}[$\\widetilde N$] is the modified period CITE{S}.", "formula-integrals": "$\\displaystyle N(t)=\\frac1\\pi\\int_0^\\pi F_t(\\theta)\\,d\\theta,\\qquad F_t(\\theta)=\\log\\max\\{|y_1|,|y_2|\\}.$ Here $y_1,y_2$ are the roots of $y^2+(x^2-kx)y+x=0$ at $x=e^{i\\theta}$ and $k=\\sqrt[3]{t}\\in\\mathbb R$. For $-1<t<27$, set $c=\\arccos((k-1)/2)\\in[0,\\pi]$, $I(t)=\\pi^{-1}\\int_0^c F_t(\\theta)\\,d\\theta$ and $J(t)=\\pi^{-1}\\int_c^\\pi F_t(\\theta)\\,d\\theta$. Then $N(t)=I(t)+J(t)$ and $\\widetilde N(t)=I(t)-2J(t)$ CITE{S}, Lemma 3. Root ties give $F_t=0$.", "formula-cm": "$\\begin{aligned}C_t:\\quad Y^2={}&X^3-27t^2X^2\\\\&+216t^3(t-27)X-432t^4(t-27)^2,\\end{aligned}\\qquad j(C_t)=\\frac{t(t+216)^3}{(t-27)^3}.$ This model is nonsingular for $t\\ne0,27$ CITE{ST}. At a selected CM argument, $H_\\Delta(j(C_t))=0$ for its imaginary quadratic order discriminant $\\Delta$; HREF{Hilbert_class_polynomials}[the Hilbert class polynomials] hold these exact polynomials.", "formula-row16": "$\\displaystyle D(729+405\\sqrt3)=\\frac{2187}{\\pi^4}L(E_{36},2)L(E_{144},2),$ where $E_{36}:y^2=x^3+1$ and $E_{144}:y^2=x^3-1$ are elliptic curves over $\\mathbb Q$ of conductors $36$ and $144$, and $L(E,s)$ is the arithmetic Hasse-Weil $L$-function CITE{TG}, Theorem 1.4 and its proof; CITE{ST}, Table 4, row 16."}, "Similar tables": [{"table": "HREF{T649}[Mahler measures of the Hessian family]", "relation": "holds the ordinary scalar periods $N(t)$ used when $t$ is outside $(-1,27)$"}, {"table": "HREF{T650}[Modified Mahler measures of the Hessian family]", "relation": "holds the signed periods $\\widetilde N(t)$ used when $-1<t<27$"}, {"table": "HREF{Hilbert_class_polynomials}", "relation": "holds the exact polynomials that characterize the CM j-invariants of the selected Hessian curves"}], "Keywords": ["Hesse", "Hessian", "Mahler measure", "Mahler determinant", "CM", "Samart", "Samart-Tao", "x^3+y^3+1-kxy", "quadratic CM determinant"], "Programs": {"program-python": {"language": "Python", "code": "# Download analytic.py from this table's attachments.\n# Its evaluator reconstructs exact radical arguments at each working precision.\nimport mpmath as mp\nfrom analytic import nu\n\ndef determinant(t, conjugate, a, b, working=140):\n    with mp.workdps(working):\n        u = mp.mpf(nu(t, working))\n        v = mp.mpf(nu(conjugate, working))\n        return a*u*u + b*v*v\n\n# Table 1, row 16. Coefficients belong to this oriented argument.\nt, conjugate = '729+405*sqrt(3)', '729-405*sqrt(3)'\nlow = determinant(t, conjugate, 4, -1, 140)\nhigh = determinant(t, conjugate, 4, -1, 190)\nwith mp.workdps(200):\n    assert abs((low-high)/high) < mp.mpf('1e-115')\n    print(mp.nstr(high, 100))", "comment": "The analytic evaluator covers the arguments in the attached catalogue. This example computes one determinant and checks agreement at two working precisions. Use the prescribed coefficients for each oriented parameter; generate.py reproduces the complete fifteen-entry selection."}}}