Quadratic CM determinants of Hessian Mahler measures

edit · history · discussion · files · long url · integral L-function special values elliptic curves number theory

Numbers

$t$

$D(t)$

$108+54\sqrt{2}$:

7.092986541447182950242847706092825572773193761643221036343250881638249829759358487071944225821788192

comment: The coefficients are $a=2$ and $b=1$ ([1], Table 1, row 13). The period at $t$ is $N$ and the period at $t^\sigma$ is $N$. Here $t=t_Q([3,0,2])$ and $t^\sigma=t_Q([6,0,1])$, with CM order discriminant $\Delta=-24$.

$18+6\sqrt{3}$:

3.215441190507797646935744825321917295105956335366857719546111742743651310674569693262700183971555823

comment: The coefficients are $a=2$ and $b=1$ ([1], Table 1, row 14). The period at $t$ is $N$ and the period at $t^\sigma$ is $\widetilde N$. Here $t=t_Q([9,0,1])$ and $t^\sigma=t_Q([9,6,2])$, with CM order discriminant $\Delta=-36$.

$270+162\sqrt{3}$:

9.646323571523392940807234475965751885317869006100573158638335228230953932023709079788100551914667469

comment: The coefficients are $a=2$ and $b=1$ ([1], Table 1, row 15). The period at $t$ is $N$ and the period at $t^\sigma$ is $N$. Here $t=t_Q([1,0,1])$ and $t^\sigma=t_Q([2,2,1])$, with CM order discriminant $\Delta=-4$.

$729+405\sqrt{3}$:

22.46796813778818642931994948020389706049816774683251140180796289188299013168800016813735310496534755

comment: The coefficients are $a=4$ and $b=-1$ ([1], Table 1, row 16). The period at $t$ is $N$ and the period at $t^\sigma$ is $N$. Here $t=t_Q([3,0,4])$ and $t^\sigma=t_Q([12,0,1])$, with CM order discriminant $\Delta=-48$.

$\frac{-27}{2}-\frac{27}{2}\sqrt{5}$:

10.72313320373494018150350977439978658466128463871421104376126607645189949646694950294938334208542441

comment: The coefficients are $a=5$ and $b=1$ ([1], Table 1, row 17). The period at $t$ is $N$ and the period at $t^\sigma$ is $\widetilde N$. Here $t=t_Q([3,-3,2])$ and $t^\sigma=t_Q([6,3,1])$, with CM order discriminant $\Delta=-15$.

$\frac{3375}{2}+\frac{1485}{2}\sqrt{5}$:

37.53096621307229063526228421039925304631449623549973865316443126758164823763432326032284169729898543

comment: The coefficients are $a=5$ and $b=1$ ([1], Table 1, row 18). The period at $t$ is $N$ and the period at $t^\sigma$ is $N$. Here $t=t_Q([3,0,5])$ and $t^\sigma=t_Q([15,0,1])$, with CM order discriminant $\Delta=-60$.

$-4320-1944\sqrt{5}$:

224.6757247845783219820084691590401519305673376921029709373504752518183988426475211259287440756456580

comment: The coefficients are $a=25$ and $b=-1$ ([1], Table 1, row 19). The period at $t$ is $N$ and the period at $t^\sigma$ is $\widetilde N$. Here $t=t_Q([3,-3,7])$ and $t^\sigma=t_Q([21,3,1])$, with CM order discriminant $\Delta=-75$.

$32+2\sqrt{6}$:

3.502588695107078646422882472997385988169702474991076464997298816933667951825228498024783819205000639

comment: The coefficients are $a=2$ and $b=1$ ([1], Table 1, row 20). The period at $t$ is $N$ and the period at $t^\sigma$ is $N$. Here $t=t_Q([9,0,2])$ and $t^\sigma=t_Q([18,0,1])$, with CM order discriminant $\Delta=-72$.

$3672+1458\sqrt{6}$:

19.84800260560677899639633401365185393296164735828276663498469329595745172700962815547377497549500362

comment: The coefficients are $a=2$ and $b=1$ ([1], Table 1, row 21). The period at $t$ is $N$ and the period at $t^\sigma$ is $N$. Here $t=t_Q([1,0,2])$ and $t^\sigma=t_Q([2,0,1])$, with CM order discriminant $\Delta=-8$.

$-864-216\sqrt{17}$:

109.1634059675396927569206469171382947552509253067376938515398601944019003292932366917923776570625801

comment: The coefficients are $a=17$ and $b=1$ ([1], Table 1, row 22). The period at $t$ is $N$ and the period at $t^\sigma$ is $\widetilde N$. Here $t=t_Q([3,-3,5])$ and $t^\sigma=t_Q([15,3,1])$, with CM order discriminant $\Delta=-51$.

$-163296-35640\sqrt{21}$:

873.9033292792856531182314015727567039171374015464944235719340742008187516874623002450580743439875301

comment: The coefficients are $a=49$ and $b=-1$ ([1], Table 1, row 23). The period at $t$ is $N$ and the period at $t^\sigma$ is $\widetilde N$. Here $t=t_Q([3,-3,13])$ and $t^\sigma=t_Q([39,3,1])$, with CM order discriminant $\Delta=-147$.

$4-4\sqrt{33}$:

16.11774496914165624216791929014262408143631171882729812301796159647007002786847706274513275220933787

comment: The coefficients are $a=11$ and $b=1$ ([1], Table 1, row 24). The period at $t$ is $N$ and the period at $t^\sigma$ is $\widetilde N$. Here $t=t_Q([9,-9,5])$ and $t^\sigma=t_Q([27,3,1])$, with CM order discriminant $\Delta=-99$.

$-16740-2916\sqrt{33}$:

134.3145414095138020180659940845218673453025976568941510251496799705839168989039755228761062684111489

comment: The coefficients are $a=11$ and $b=1$ ([1], Table 1, row 25). The period at $t$ is $N$ and the period at $t^\sigma$ is $\widetilde N$. Here $t=t_Q([1,-1,3])$ and $t^\sigma=t_Q([5,3,1])$, with CM order discriminant $\Delta=-11$.

$-55296-8640\sqrt{41}$:

618.2208359419764200914664915370125555404604865786987689205508265792866246770171059802078995552867217

comment: The coefficients are $a=41$ and $b=1$ ([1], Table 1, row 26). The period at $t$ is $N$ and the period at $t^\sigma$ is $\widetilde N$. Here $t=t_Q([3,-3,11])$ and $t^\sigma=t_Q([33,3,1])$, with CM order discriminant $\Delta=-123$.

$-13500000-1431000\sqrt{89}$:

2899.206750670118464346726431718974939988615770205808065544853280436864224226453978998417683054618882

comment: The coefficients are $a=89$ and $b=1$ ([1], Table 1, row 27). The period at $t$ is $N$ and the period at $t^\sigma$ is $\widetilde N$. Here $t=t_Q([3,-3,23])$ and $t^\sigma=t_Q([69,3,1])$, with CM order discriminant $\Delta=-267$.

Definition

The quadratic CM Mahler determinant [1] is $D(t)=a\nu(t)^2+b\nu(t^\sigma)^2$, with the oriented parameter and fixed coefficients in (1).

Parameters

$t$

— oriented Hessian CM parameter ($t$ is a real quadratic CM parameter of $C_t$, with its embedding and determinant normalization fixed as in [1].)

Formulas

(1)

$D(t)=\det\begin{pmatrix}a\nu(t)&b\nu(t^\sigma)\\-\nu(t^\sigma)&\nu(t)\end{pmatrix}=a\nu(t)^2+b\nu(t^\sigma)^2.$ Here $t^\sigma$ is the algebraic conjugate of the oriented real quadratic CM argument $t$, $a,b$ are its fixed normalization coefficients from [1], Table 1, and $\nu$ is the period in (2). This is the normalization of [1], equation (5.5).

(2)

$\displaystyle \nu(t)=\begin{cases}\widetilde N(t),&-1<t<27,\\N(t),&t\leq-1\ \text{or}\ t\geq27.\end{cases}$ Here $N$ is the ordinary Hessian Mahler measure and $\widetilde N$ is the modified period [2].

(3)

$\displaystyle N(t)=\frac1\pi\int_0^\pi F_t(\theta)\,d\theta,\qquad F_t(\theta)=\log\max\{|y_1|,|y_2|\}.$ 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 $\widetilde N(t)=I(t)-2J(t)$ [2], Lemma 3. Root ties give $F_t=0$.

(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 $\Delta$ [1], 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)

$\displaystyle D(729+405\sqrt3)=\frac{2187}{\pi^4}L(E_{36},2)L(E_{144},2),$ 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 [3], Theorem 1.4 and its proof; [1], Table 4, row 16.

(6)

$\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 [1], Sections 2 and 3.

Comments

(7)

The square roots in the parameter labels are positive. Each entry gives its prescribed integers $a,b$ and source row in [1], Table 1. They are fixed normalization data, not independently variable arguments. The determinant retains its sign; no absolute value or primitive rescaling is applied.

(8)

The ordinary logarithmic Mahler measure [4] $N(t)$ and the modified period $\widetilde N(t)$ 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 (2).

(9)

The arguments are the fifteen oriented real quadratic CM representatives in [1], Table 1, rows 13 through 27. Each orientation and its coefficients are retained: the listed representative is not always the smaller conjugate. The degree is that of $t$, not $\sqrt[3]{t}$. Four-dimensional determinants of totally real quartic CM orbits are a separate family, not missing rows of this table.

(10)

The determinant normalization comes from the arithmetic identities of [1], Theorem 5.2. The special case $t=729+405\sqrt3$ also appears in [3], Theorem 1.4, whose scalar convention is $m_3=3N$; its two-dimensional determinant is therefore nine times $D(t)$.

(11)

In $t_Q([A,B,C])$, the triple specifies the CM point $\tau_Q([A,B,C])$ in (6), 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.

Programs

(P1)

Sage

from generate import Table

table = Table()
print(table.value({'t': '108+54*sqrt(2)'}, 100)['number'])

References

[1]

D. Samart and Z. Tao, Determinants of Mahler measures and special values of L-functions, version 3 (2026). (arXiv)

[2]

D. Samart, Mahler measure of a nonreciprocal family of elliptic curves, corrected version 3 (2023). (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

[4]

Mahler measure

Similar tables

Mahler measures of the Hessian family — holds the ordinary scalar periods $N(t)$ used when $t$ is outside $(-1,27)$

Modified Mahler measures of the Hessian family — holds the signed periods $\widetilde N(t)$ used when $-1<t<27$

Hilbert class polynomials — holds the exact polynomials that characterize the CM j-invariants of the selected Hessian curves

Data properties

Entries are of type: real number

How well the digits are known: heuristic (agreement-checked)

Table is complete: yes (it holds all fifteen real quadratic CM orbits classified exhaustively in [1], Proposition 3.4 and Section 3, with one oriented representative and its prescribed coefficients from Theorem 5.2 and Table 1, rows 13 through 27; this completeness concerns the specified quadratic determinants, not arbitrary coefficients or higher-degree orbits)

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.

All 15 entries were recomputed; nonzero values agree at the two nominal precisions to at least 131.39 relative decimal digits. The table retains 100 significant digits.

All fifteen prescribed quadratic combinations were separately reconstructed from the independently computed scalar periods using decimal arithmetic. They agree with the main determinants to over 110 relative digits. Every pre-existing numeric payload is retained unchanged.

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.