Quartic CM determinants of Hessian Mahler measures

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Numbers

$t$

$D_4(t)$

$t_Q([27,24,8])$:

1239.286015777256461510332910345458008585469717572054316701599878327177129771518264424577531013089010

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-288$. The prescribed normalization is $(a,b,c,d)=(2,1,1,8)$, with sign variant $j=4$ ([1], Tables 2 and 3, row 52). The scalar inputs are $\widetilde N(t_1)$ at $t_1=t_Q([27,24,8])$; $\widetilde N(t_2)$ at $t_2=t_Q([27,12,4])$; $N(t_3)$ at $t_3=t_Q([72,0,1])$; $N(t_4)$ at $t_4=t_Q([9,0,8])$.

$t_Q([12,12,5])$:

8849.466634150676536532326333968866425243054436955746294300631419086614585963613892000578453182079633

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-96$. The prescribed normalization is $(a,b,c,d)=(2,1,1,8)$, with sign variant $j=4$ ([1], Tables 2 and 3, row 53). The scalar inputs are $N(t_1)$ at $t_1=t_Q([12,12,5])$; $\widetilde N(t_2)$ at $t_2=t_Q([15,12,4])$; $N(t_3)$ at $t_3=t_Q([24,0,1])$; $N(t_4)$ at $t_4=t_Q([3,0,8])$.

$t_Q([12,-12,7])$:

134701.8642129649650863185446240623187281384231882552344197683961239279293939928934702872523063694296

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-192$. The prescribed normalization is $(a,b,c,d)=(4,1,1,16)$, with sign variant $j=6$ ([1], Tables 2 and 3, row 54). The scalar inputs are $N(t_1)$ at $t_1=t_Q([12,-12,7])$; $\widetilde N(t_2)$ at $t_2=t_Q([21,12,4])$; $N(t_3)$ at $t_3=t_Q([48,0,1])$; $N(t_4)$ at $t_4=t_Q([3,0,16])$.

$t_Q([4,-4,3])$:

77248.82831678231943414408474486688253516094572865805240773305908239404108909130514913199943314921493

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-32$. The prescribed normalization is $(a,b,c,d)=(2,1,1,8)$, with sign variant $j=4$ ([1], Tables 2 and 3, row 55). The scalar inputs are $N(t_1)$ at $t_1=t_Q([4,-4,3])$; $\widetilde N(t_2)$ at $t_2=t_Q([8,8,3])$; $N(t_3)$ at $t_3=t_Q([8,0,1])$; $N(t_4)$ at $t_4=t_Q([1,0,8])$.

$t_Q([30,0,1])$:

28007.16671996294368772676027572503496320341630405352902967719323920751260891522203937602111963050834

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-120$. The prescribed normalization is $(a,b,c,d)=(1,2,5,10)$, with sign variant $j=4$ ([1], Tables 2 and 3, row 56). The scalar inputs are $N(t_1)$ at $t_1=t_Q([30,0,1])$; $N(t_2)$ at $t_2=t_Q([15,0,2])$; $N(t_3)$ at $t_3=t_Q([6,0,5])$; $N(t_4)$ at $t_4=t_Q([3,0,10])$.

$t_Q([18,18,7])$:

529.1099380834092318404369804098330739562864169919158998000940145246520666701214476501189448143185242

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-180$. The prescribed normalization is $(a,b,c,d)=(5,1,2,10)$, with sign variant $j=2$ ([1], Tables 2 and 3, row 57). The scalar inputs are $\widetilde N(t_1)$ at $t_1=t_Q([18,18,7])$; $\widetilde N(t_2)$ at $t_2=t_Q([27,6,2])$; $N(t_3)$ at $t_3=t_Q([45,0,1])$; $N(t_4)$ at $t_4=t_Q([9,0,5])$.

$t_Q([2,-2,3])$:

61200.38283831433448287721073407069222094379556539827241021087434668475571151071411153042461685617597

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-20$. The prescribed normalization is $(a,b,c,d)=(5,1,2,10)$, with sign variant $j=2$ ([1], Tables 2 and 3, row 58). The scalar inputs are $N(t_1)$ at $t_1=t_Q([2,-2,3])$; $\widetilde N(t_2)$ at $t_2=t_Q([7,6,2])$; $N(t_3)$ at $t_3=t_Q([5,0,1])$; $N(t_4)$ at $t_4=t_Q([1,0,5])$.

$t_Q([60,0,1])$:

337154.2409054343500552887850263443846265378751771029214171815764103107717929289958629989777716195982

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-240$. The prescribed normalization is $(a,b,c,d)=(1,4,5,20)$, with sign variant $j=1$ ([1], Tables 2 and 3, row 59). The scalar inputs are $N(t_1)$ at $t_1=t_Q([60,0,1])$; $N(t_2)$ at $t_2=t_Q([15,0,4])$; $N(t_3)$ at $t_3=t_Q([12,0,5])$; $N(t_4)$ at $t_4=t_Q([3,0,20])$.

$t_Q([6,-6,5])$:

27026.37700300738176357349616404517809102519303463606927911303740746186531496876686400844714159423392

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-84$. The prescribed normalization is $(a,b,c,d)=(7,1,2,14)$, with sign variant $j=1$ ([1], Tables 2 and 3, row 60). The scalar inputs are $N(t_1)$ at $t_1=t_Q([6,-6,5])$; $\widetilde N(t_2)$ at $t_2=t_Q([15,6,2])$; $N(t_3)$ at $t_3=t_Q([21,0,1])$; $N(t_4)$ at $t_4=t_Q([3,0,7])$.

$t_Q([6,-6,7])$:

154737.0272540944066501680425011729739037648941562753861595438601458528299135513389388667216061105604

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-132$. The prescribed normalization is $(a,b,c,d)=(11,1,2,22)$, with sign variant $j=2$ ([1], Tables 2 and 3, row 61). The scalar inputs are $N(t_1)$ at $t_1=t_Q([6,-6,7])$; $\widetilde N(t_2)$ at $t_2=t_Q([21,6,2])$; $N(t_3)$ at $t_3=t_Q([33,0,1])$; $N(t_4)$ at $t_4=t_Q([3,0,11])$.

$t_Q([78,0,1])$:

1231184.020139706764901086152974608240035405127501761681378923208864895009977288937949474469450262643

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-312$. The prescribed normalization is $(a,b,c,d)=(1,2,13,26)$, with sign variant $j=3$ ([1], Tables 2 and 3, row 62). The scalar inputs are $N(t_1)$ at $t_1=t_Q([78,0,1])$; $N(t_2)$ at $t_2=t_Q([39,0,2])$; $N(t_3)$ at $t_3=t_Q([6,0,13])$; $N(t_4)$ at $t_4=t_Q([3,0,26])$.

$t_Q([3,-3,17])$:

2404722.467075383055619967516386067136452200015849867328743276871177125043876336938322156968379820891

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-195$. The prescribed normalization is $(a,b,c,d)=(65,13,5,1)$, with sign variant $j=3$ ([1], Tables 2 and 3, row 63). The scalar inputs are $N(t_1)$ at $t_1=t_Q([3,-3,17])$; $N(t_2)$ at $t_2=t_Q([15,15,7])$; $\widetilde N(t_3)$ at $t_3=t_Q([21,15,5])$; $\widetilde N(t_4)$ at $t_4=t_Q([51,3,1])$.

$t_Q([102,0,1])$:

3625054.428240183932133514810571940689323968894541849433295019496820557755123989982523916406475341055

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-408$. The prescribed normalization is $(a,b,c,d)=(1,2,17,34)$, with sign variant $j=4$ ([1], Tables 2 and 3, row 64). The scalar inputs are $N(t_1)$ at $t_1=t_Q([102,0,1])$; $N(t_2)$ at $t_2=t_Q([51,0,2])$; $N(t_3)$ at $t_3=t_Q([6,0,17])$; $N(t_4)$ at $t_4=t_Q([3,0,34])$.

$t_Q([42,0,1])$:

102407.6172638596961800410751935745422577815936307435078122700113605794128679417073359667599725967379

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-168$. The prescribed normalization is $(a,b,c,d)=(1,2,7,14)$, with sign variant $j=3$ ([1], Tables 2 and 3, row 65). The scalar inputs are $N(t_1)$ at $t_1=t_Q([42,0,1])$; $N(t_2)$ at $t_2=t_Q([21,0,2])$; $N(t_3)$ at $t_3=t_Q([6,0,7])$; $N(t_4)$ at $t_4=t_Q([3,0,14])$.

$t_Q([6,-6,11])$:

1421481.283625590309692634374172251019402255991470903779421220881537112382904192420597925349524212589

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-228$. The prescribed normalization is $(a,b,c,d)=(19,1,2,38)$, with sign variant $j=1$ ([1], Tables 2 and 3, row 66). The scalar inputs are $N(t_1)$ at $t_1=t_Q([6,-6,11])$; $\widetilde N(t_2)$ at $t_2=t_Q([33,6,2])$; $N(t_3)$ at $t_3=t_Q([57,0,1])$; $N(t_4)$ at $t_4=t_Q([3,0,19])$.

$t_Q([9,-9,11])$:

22921.63793849829383419022546404339159202210511757662011481457682223529915163225206613403556136007194

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-315$. The prescribed normalization is $(a,b,c,d)=(35,7,5,1)$, with sign variant $j=3$ ([1], Tables 2 and 3, row 67). The scalar inputs are $N(t_1)$ at $t_1=t_Q([9,-9,11])$; $\widetilde N(t_2)$ at $t_2=t_Q([27,21,7])$; $\widetilde N(t_3)$ at $t_3=t_Q([27,15,5])$; $\widetilde N(t_4)$ at $t_4=t_Q([81,3,1])$.

$t_Q([1,-1,9])$:

1836277.883739696650494572506619476148649770865530304789197923320981294520925205971075848848860067986

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-35$. The prescribed normalization is $(a,b,c,d)=(35,7,5,1)$, with sign variant $j=3$ ([1], Tables 2 and 3, row 68). The scalar inputs are $N(t_1)$ at $t_1=t_Q([1,-1,9])$; $N(t_2)$ at $t_2=t_Q([5,-5,3])$; $N(t_3)$ at $t_3=t_Q([7,7,3])$; $\widetilde N(t_4)$ at $t_4=t_Q([11,3,1])$.

$t_Q([3,-3,37])$:

60142784.02536301452058225133984876004659445276096447106375427972469441818594894645828413867293815366

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-435$. The prescribed normalization is $(a,b,c,d)=(145,29,5,1)$, with sign variant $j=4$ ([1], Tables 2 and 3, row 69). The scalar inputs are $N(t_1)$ at $t_1=t_Q([3,-3,37])$; $N(t_2)$ at $t_2=t_Q([15,-15,11])$; $\widetilde N(t_3)$ at $t_3=t_Q([33,15,5])$; $\widetilde N(t_4)$ at $t_4=t_Q([111,3,1])$.

$t_Q([6,-6,17])$:

10025640.62159877856653785959482570635939739282707522458192667239880553638408870745968781468902458916

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-372$. The prescribed normalization is $(a,b,c,d)=(31,1,2,62)$, with sign variant $j=1$ ([1], Tables 2 and 3, row 70). The scalar inputs are $N(t_1)$ at $t_1=t_Q([6,-6,17])$; $\widetilde N(t_2)$ at $t_2=t_Q([51,6,2])$; $N(t_3)$ at $t_3=t_Q([93,0,1])$; $N(t_4)$ at $t_4=t_Q([3,0,31])$.

$t_Q([3,-3,47])$:

158620343.0762330030264350086559565533739198316581592848773041010984362699482460989927635199510711093

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-555$. The prescribed normalization is $(a,b,c,d)=(185,37,5,1)$, with sign variant $j=3$ ([1], Tables 2 and 3, row 71). The scalar inputs are $N(t_1)$ at $t_1=t_Q([3,-3,47])$; $N(t_2)$ at $t_2=t_Q([15,-15,13])$; $\widetilde N(t_3)$ at $t_3=t_Q([39,15,5])$; $\widetilde N(t_4)$ at $t_4=t_Q([141,3,1])$.

$t_Q([3,-3,67])$:

670274151.1245003192869603414011225872602981327427100679734922770126296086144445401725312397447525861

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-795$. The prescribed normalization is $(a,b,c,d)=(265,53,5,1)$, with sign variant $j=4$ ([1], Tables 2 and 3, row 72). The scalar inputs are $N(t_1)$ at $t_1=t_Q([3,-3,67])$; $N(t_2)$ at $t_2=t_Q([15,-15,17])$; $\widetilde N(t_3)$ at $t_3=t_Q([51,15,5])$; $\widetilde N(t_4)$ at $t_4=t_Q([201,3,1])$.

$t_Q([6,-6,31])$:

131039054.3095260405487443949342159141633029150228693035786853205156246627694146631076028318744837274

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-708$. The prescribed normalization is $(a,b,c,d)=(59,1,2,118)$, with sign variant $j=2$ ([1], Tables 2 and 3, row 73). The scalar inputs are $N(t_1)$ at $t_1=t_Q([6,-6,31])$; $\widetilde N(t_2)$ at $t_2=t_Q([93,6,2])$; $N(t_3)$ at $t_3=t_Q([177,0,1])$; $N(t_4)$ at $t_4=t_Q([3,0,59])$.

$t_Q([3,-3,41])$:

89597695.21574711373314389355191936941093563833159184152781159728233292217908776557264758144810324501

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-483$. The prescribed normalization is $(a,b,c,d)=(161,23,7,1)$, with sign variant $j=1$ ([1], Tables 2 and 3, row 74). The scalar inputs are $N(t_1)$ at $t_1=t_Q([3,-3,41])$; $N(t_2)$ at $t_2=t_Q([21,-21,11])$; $\widetilde N(t_3)$ at $t_3=t_Q([33,21,7])$; $\widetilde N(t_4)$ at $t_4=t_Q([123,3,1])$.

$t_Q([3,-3,53])$:

255099672.9560620339045592887388675540046216849968110275021606665970274539206884085734751845483737806

comment: The ordered inputs $t_1<t_2<t_3<t_4$ have CM order discriminant $\Delta=-627$. The prescribed normalization is $(a,b,c,d)=(209,19,11,1)$, with sign variant $j=3$ ([1], Tables 2 and 3, row 75). The scalar inputs are $N(t_1)$ at $t_1=t_Q([3,-3,53])$; $N(t_2)$ at $t_2=t_Q([33,33,13])$; $\widetilde N(t_3)$ at $t_3=t_Q([39,33,11])$; $\widetilde N(t_4)$ at $t_4=t_Q([159,3,1])$.

Definition

At a totally real quartic Hessian CM parameter $t$, the Mahler determinant is $D_4(t)=\det M_j$, with the conjugate periods and signed matrix specified in (1) and (2).

Parameters

$t$

— least conjugate of the Hessian CM orbit ($t$ is the least real conjugate of a totally real quartic algebraic parameter for which $C_t$ has complex multiplication, with the determinant normalization prescribed in [1], Tables 2 and 3.)

Formulas

(1)

$\displaystyle B=\begin{pmatrix}w_1&w_2&w_3&w_4\\w_2&w_1&w_4&w_3\\w_3&w_4&w_1&w_2\\w_4&w_3&w_2&w_1\end{pmatrix},\qquad (w_1,w_2,w_3,w_4)=(\sqrt a\,\nu(t_1),\sqrt b\,\nu(t_2),\sqrt c\,\nu(t_3),\sqrt d\,\nu(t_4)),$ and $M_j=S_j\circ B$, where $\circ$ denotes entrywise multiplication. Thus $D_4(t)=\det M_j$ is equation (5.10) of [1] with $\mu$ replaced by $\nu$ and with the signs from equation (5.11).

(2)

$\displaystyle S_1=\begin{pmatrix}+1&+1&+1&+1\\+1&+1&+1&+1\\+1&+1&-1&-1\\+1&+1&-1&-1\end{pmatrix},\quad S_2=\begin{pmatrix}+1&+1&+1&+1\\+1&-1&-1&+1\\+1&-1&+1&-1\\+1&+1&-1&-1\end{pmatrix},$
$\displaystyle S_3=\begin{pmatrix}+1&+1&+1&+1\\+1&-1&+1&-1\\+1&+1&+1&+1\\+1&-1&+1&-1\end{pmatrix},\quad S_4=\begin{pmatrix}+1&+1&+1&+1\\+1&-1&+1&-1\\+1&+1&-1&-1\\+1&-1&-1&+1\end{pmatrix},$
$\displaystyle S_5=\begin{pmatrix}+1&+1&+1&+1\\+1&-1&-1&+1\\+1&-1&-1&+1\\+1&+1&+1&+1\end{pmatrix},\quad S_6=\begin{pmatrix}+1&+1&+1&+1\\+1&+1&-1&-1\\+1&-1&+1&-1\\+1&-1&-1&+1\end{pmatrix}.$ These are the six entrywise sign patterns of [1], equation (5.11); the row's prescribed variant is fixed.

(3)

$\displaystyle \nu(u)=\begin{cases}\widetilde N(u),&-1<u<27,\\N(u),&u\leq-1\text{ or }u\geq27.\end{cases}$ Here $N(u)=m(x^3+y^3+1-\sqrt[3]{u}xy)$ and the real cube root is used [1], Sections 1 and 2.

(4)

$\displaystyle F_u(\theta)=\log\max\{|y_1|,|y_2|\},\qquad N(u)=\frac1\pi\int_0^\pi F_u(\theta)\,d\theta,$ where $y_1,y_2$ solve $y^2+(x^2-kx)y+x=0$ at $x=e^{i\theta}$ and $k=\sqrt[3]{u}\in\mathbb R$. For $-1<u<27$, put $c(u)=\arccos((k-1)/2)$ and define $I(u)=\pi^{-1}\int_0^{c(u)}F_u(\theta)\,d\theta$ and $J(u)=\pi^{-1}\int_{c(u)}^\pi F_u(\theta)\,d\theta$. Then $N(u)=I(u)+J(u)$ and $\widetilde N(u)=I(u)-2J(u)$ [2]. Since $y_1y_2=x$, ties in root modulus give $F_u=0$.

(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 [1], Sections 2 and 3.

(6)

$\displaystyle C_u:\ Y^2=X^3-27u^2X^2+216u^3(u-27)X-432u^4(u-27)^2,\qquad j(C_u)=\frac{u(u+216)^3}{(u-27)^3}.$ The model is nonsingular for $u\ne0,27$. For $u=t_Q([A,B,C])$, its $j$-invariant is $j(\tau_Q([A,B,C]))$ and satisfies $H_\Delta(j(C_u))=0$, 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 covers some of these orders, not necessarily every discriminant used here.

(7)

$\displaystyle D_4(t)=\frac{r}{\pi^8}L(C_t/K,2),\qquad K=\mathbb Q(t),$ with the fixed rational $r$ in [1], Table 3 and Theorem 5.4. The $L$-function is the arithmetic Hasse-Weil $L$-function over the degree-four field $K$, not the $L$-function of a rational curve with the same $j$-invariant.

Comments

(8)

Every entry specifies the positive integers $(a,b,c,d)$ and the sign variant $j$ attached to its ordered orbit. The weights are the positive square roots $\sqrt a,\sqrt b,\sqrt c,\sqrt d$, not $a,b,c,d$. The determinant is stored with its sign and published scale, without taking an absolute value or dividing out a common factor. The coefficients and sign variant are assigned to the orbit in [1], Tables 2 and 3; they are not additional free parameters.

(9)

The scalar input is the piecewise period $\nu$: it uses the ordinary logarithmic Mahler measure $N$ outside $(-1,27)$ and the signed modified period $\widetilde N$ inside that interval. Both use natural logarithms and the real cube root of the Hessian parameter. The ordinary measure and modified period are different quantities, not interchangeable numerical implementations.

(10)

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. Individual CM representatives and algebraic expressions are supplied in [3].

(11)

$t=t_1<t_2<t_3<t_4$ orders the four real conjugates. There is one determinant per Galois orbit, indexed by its least conjugate. Its argument key is the same as that scalar argument in T649. The other three conjugates occur as scalar inputs and links in the entry comment, not as duplicate determinant entries. Reordering conjugates without transporting the weights and signs changes the definition.

(12)

The degree is $[\mathbb Q(t):\mathbb Q]=4$, not the degree of $k=\sqrt[3]{t}$ or the class number of the CM order. The source's totally real quartic orbits are biquadratic. Real arguments in non-totally-real cubic or quartic orbits can belong in the scalar tables, but do not define entries of this totally real quartic determinant family.

(13)

These determinants occur in the special-value identities of [1], Theorem 5.4. They are signed quantities with a specified matrix normalization, not a claim that a basis of the integral regulator lattice has been independently constructed or that a general case of Beilinson's conjecture has been proved here.

Programs

(P1)

Sage

from generate import Table

table = Table()
print(table.value({'t': 'cm-m288-27-24-8'}, 100)['number'])

References

[1]

D. Samart and Z. Tao, Determinants of Mahler measures and special values of L-functions, version 3 (2026), Sections 2, 3 and 5, Tables 2 and 3. (arXiv)

[2]

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

Links

[3]

Samart and Tao, supplementary CM parameters and special-value formulas

Similar tables

Mahler measures of the Hessian family — holds the ordinary scalar periods at the four conjugate arguments; these are the determinant inputs only outside the open interval $(-1,27)$

Modified Mahler measures of the Hessian family — holds the signed scalar periods used by this determinant at conjugates in $(-1,27)$

Quadratic CM determinants of Hessian Mahler measures — holds the corresponding two-dimensional determinants at real quadratic CM orbits, with their separate prescribed normalization

Hilbert class polynomials — characterizes the CM orders used to identify the Hessian parameters; the level-three form additionally specifies the parameter and embedding

Data properties

Entries are of type: real number

Table is complete: yes (it holds all 24 totally real quartic CM orbits in the classification of [1], Section 3 and Table 2, rows 52 through 75, with the fixed determinant normalizations from Table 3 and Theorem 5.4; this concerns the specified signed determinants, not arbitrary weights or nonreal conjugates)

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 24 entries were recomputed; nonzero values agree at the two nominal precisions to at least 131.01 relative decimal digits. The table retains 100 significant digits.

Independent scalar periods at nominal precisions of 125 and 145 digits, with extra arithmetic digits recorded for each input, and a separately implemented signed determinant agree with all 24 results to over 110 relative digits. Separate exact polynomial expansions check all sign variants and weights. The largest measured absolute-term sum divided by determinant magnitude is less than 3. The quartic Hasse-Weil L-functions were not independently evaluated here.

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.

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