\begin{info}
\section{Results for the cubic $L_4$}
We consider the following cubic
\begin{equation}\tag{$L_4$}X_0^3+X_1^3+2X_2^3+2X_3^3=0
\end{equation}
\subsection{The Galois group}
The splitting Galois group for the
cubic is given by
\[G=\langle c^1,\tau_2^{1}\tau_3^{1}\rangle.
\]
\subsection{The Picard group}The orbits of the 27 lines are given by
\[\begin{split}
O_{0}&=\{L(0,0)\},\\
O_{1}&=\{L(1,0),L(2,0)\},\\
O_{2}&=\{L(0,1),L(0,2)\},\\
O_{3}&=\{L(1,1),L(2,2)\},\\
O_{4}&=\{L(2,1),L(1,2)\},\\
O_{5}&=\{M(0,0),M(0,1),M(1,0),M(2,0),M(1,1),M(2,1)\},\\
O_{6}&=\{M(0,2),M(1,2),M(2,2)\},\\
O_{7}&=\{N(0,0),N(0,2),N(1,0),N(2,0),N(1,2),N(2,2)\},\\
O_{8}&=\{N(0,1),N(1,1),N(2,1)\}.
\end{split}\]
In order to describe
the Picard group, we consider the cubic over $\overline {Q}$
as the projective plane blown up in six points
so that the $27$ lines are given by
\begin{align*}
E_1&=L(0,0),&E_2&=L(1,0),&E_3&=L(2,0),\\
E_4&=M(0,0),&E_5&=M(1,0),&E_6&=M(2,0),\\
Q_1&=L(1,1),&Q_2&=L(2,1),&Q_3&=L(0,1),\\
Q_4&=M(2,2),&Q_5&=M(0,2),&Q_6&=M(1,2),\\
R_{1,2}&=L(1,2),&R_{2,3}&=L(2,2),&R_{3,1}&=L(0,2)\\
R_{4,5}&=M(0,1),&R_{5,6}&=M(1,1),&R_{6,4}&=M(2,1)\\
R_{1,4}&=N(2,1),&R_{1,5}&=N(0,1),&R_{1,6}&=N(1,1)\\
R_{2,4}&=N(0,2),&R_{2,5}&=N(1,2),&R_{2,6}&=N(2,2)\\
R_{3,4}&=N(1,0),&R_{3,5}&=N(2,0),&R_{3,6}&=N(0,0).
\end{align*}
where the $E_i$ are the exceptional lines,
the $R_{i,j}$ is the strict lifting of the line $(P_iP_j)$
and $Q_i$ the strict lifting of the conic going through all points
except $P_i$. In the basis given by the lifting
$\Lambda$ of line not going through any of the six points blown up
and by the $6$ exceptional lines $E_i$, the Picard group
of the variety over $\mathbf Q$ is given by the matrix
\[\begin{pmatrix}
0&0&-3\\
1&0&0\\
0&1&0\\
0&1&0\\
0&0&1\\
0&0&1\\
0&0&1\\
\end{pmatrix}
\]
The effective cone is generated by

\[\begin{split}
[O_{1}]&= e_{0},\\
[O_{2}]&= e_{1},\\
[O_{3}]&=-2e_{0}-e_{1}-e_{2},\\
[O_{4}]&=-2e_{1}-e_{2},\\
[O_{5}]&=-2e_{0}-e_{1}-e_{2},\\
[O_{6}]&=-e_{2},\\
[O_{7}]&=-3e_{0}-3e_{1}-2e_{2},\\
[O_{8}]&=-3e_{1}-2e_{2},\\
[O_{9}]&=-3e_{0}-e_{2}.\\
\noalign{and the canonical sheaf is given by}\omega_V^{-1}&=-e_{0}-e_{1}-e_{2}.
\end{split}\]
The corresponding polyhedron may be described as
\[\begin{split}
X_1 &> 0,\\
X_2 &> 0,\\
X_1-X_2 &> -1,\\
-X_1-X_2 &> -2,\\
-2X_1+X_2 &> -1.
\end{split}\]
with $ 1$ as coefficient.
Thus the constant $\alpha(V)$ is $ 1$.

\subsection{The bad places}
\par
At the bad place $2$
We use the following reduced form

\[X_0^3+X_1^3+2X_2^3+2X_3^3=0
\]

of type $(1,a,bp,cp)$. Thus
the density at $p$ is $ 3/4$
and the zeta factor at $p$ is $ 9/16$

\par
At the bad place $3$
We use the following reduced form

\[X_0^3+X_1^3+2X_2^3+2X_3^3=0
.\]

And a direct computation gives that
the density at $p$ is $ 4/3$
and the zeta factor at $p$ is $ 4/9$

\subsection{Experimental curve}
The following graph represents the numerical data

\centerline{%
{\setlength{\unitlength}{0.240900pt}
\begin{picture}(600,900)(0,0)
\put(50.000,88.000){\rule[-0.200pt]{120.450pt}{0.400pt}}\put(50.000,88.000){\rule[-0.200pt]{0.400pt}{192.720pt}}\put(50.000,88.000){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(50.000,47.0){\makebox(0,0)[c]{0}}
\put(91.692,88.000){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(91.692,47.0){\makebox(0,0)[c]{1}}
\put(133.385,88.000){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(133.385,47.0){\makebox(0,0)[c]{2}}
\put(175.077,88.000){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(175.077,47.0){\makebox(0,0)[c]{3}}
\put(216.769,88.000){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(216.769,47.0){\makebox(0,0)[c]{4}}
\put(258.461,88.000){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(258.461,47.0){\makebox(0,0)[c]{5}}
\put(300.154,88.000){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(300.154,47.0){\makebox(0,0)[c]{6}}
\put(341.846,88.000){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(341.846,47.0){\makebox(0,0)[c]{7}}
\put(383.538,88.000){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(383.538,47.0){\makebox(0,0)[c]{8}}
\put(425.230,88.000){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(425.230,47.0){\makebox(0,0)[c]{9}}
\put(466.923,88.000){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(466.923,47.0){\makebox(0,0)[c]{10}}
\put(508.615,88.000){\rule[-0.200pt]{0.400pt}{4.818pt}}
\put(508.615,47.0){\makebox(0,0)[c]{11}}
\put(550.000,0.0){\makebox(0,0)[c]{$\log(B)$}}
\put(133.601,868.000){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(153.601,848.000){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(140.028,680.748){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(160.028,660.748){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(145.596,706.289){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(165.596,686.289){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(152.760,845.125){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(172.760,825.125){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(158.873,739.212){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(178.873,719.212){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(165.837,782.355){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(185.837,762.355){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(173.171,778.655){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(193.171,758.655){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(180.547,724.835){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(200.547,704.835){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(187.771,672.157){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(207.771,652.157){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(194.736,608.818){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(214.736,588.818){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(202.070,609.353){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(222.070,589.353){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(209.446,596.146){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(229.446,576.146){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(216.670,586.331){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(236.670,566.331){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(224.034,565.503){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(244.034,545.503){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(231.636,549.377){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(251.636,529.377){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(239.182,559.351){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(259.182,539.351){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(246.737,547.571){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(266.737,527.571){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(254.300,527.281){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(274.300,507.281){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(261.837,519.716){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(281.837,499.716){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(269.439,517.512){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(289.439,497.512){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(276.996,512.318){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(296.996,492.318){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(284.523,517.784){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(304.523,497.784){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(292.078,508.753){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(312.078,488.753){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(299.653,505.904){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(319.653,485.904){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(307.211,503.604){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(327.211,483.604){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(314.795,502.591){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(334.795,482.591){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(322.389,493.108){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(342.389,473.108){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(329.984,486.515){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(349.984,466.515){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(337.569,481.777){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(357.569,461.777){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(345.153,481.556){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(365.153,461.556){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(352.744,477.262){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(372.744,457.262){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(360.342,474.276){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(380.342,454.276){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(367.941,472.408){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(387.941,452.408){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(375.536,467.767){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(395.536,447.767){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(383.138,463.191){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(403.138,443.191){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(390.739,461.304){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(410.739,441.304){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(398.336,458.796){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(418.336,438.796){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(405.934,457.111){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(425.934,437.111){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(413.535,456.226){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(433.535,436.226){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(421.133,454.995){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(441.133,434.995){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(428.734,453.223){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(448.734,433.223){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(436.333,450.563){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(456.333,430.563){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(443.934,448.795){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(463.934,428.795){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(451.535,447.637){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(471.535,427.637){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(459.135,446.529){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(479.135,426.529){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(466.736,445.165){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(486.736,425.165){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(474.337,443.587){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(494.337,423.587){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(481.938,442.610){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(501.938,422.610){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(489.539,441.538){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(509.539,421.538){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(497.140,440.174){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(517.140,420.174){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(504.741,439.205){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(524.741,419.205){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(510.000,438.116){\rule[-0.200pt]{9.636pt}{0.400pt}}
\put(530.000,418.116){\rule[-0.200pt]{0.400pt}{9.636pt}}
\put(50.0,376.339){\rule[-0.200pt]{120.450pt}{0.400pt}}
\put(40.0,376.339){\makebox(0,0)[r]{$\thetaH(V)$}}
\end{picture}}
}
\end{info}
\ifx\FirstDAXcolumn\undefined
\createnewtable{DAX}{23}
\forDAX
\ifx\showallcolumns\undefined
\def\showallcolumns{}\fi
\gaddtomacro\showallcolumns{\showDAXcolumns}
\fi
\FirstDAXcolumn
{\text{Surface}}%
{H}%
{n_\greeksubscript{U,\mathbf H}(B)}%
{C_{\Br}}%
{\alpha(V)}%
{H^1(\mathbf Q,\Pic(\overline V))}%
{q_0}%
{\zeta^*_{\mathbf Q(q_0^{1/3})}(1)}%
{q_1}%
{\zeta^*_{\mathbf Q(q_1^{1/3})}(1)}%
{q_2}%
{\zeta^*_{\mathbf Q(q_2^{1/3})}(1)}%
{\lambda'_{3}\omegaH(V(\mathbf Q_{3}))}%
{p_{0}}%
{\lambda'_{p_{0}}\omegaH(V(\mathbf Q_{p_{0}}))}%
{C_0}%
{C_1}%
{C_2}%
{C_3}%
{\omegaH(V(\mathbf R))}%
{\thetaH(V)}%
{n_\greeksubscript{U,\mathbf H}(B)/\thetaH(V)B\log(B)^{2}}%
{\thetaH^{\mathrm{stat}}(V)/\thetaH(V)}%
\addDAXcolumn
{L_4}%
{100000}%
{3051198}%
{1}%
{1}%
{1}%
{1/4}%
{8.146241\times10^{-1}}%
{1}%
{6.045998\times10^{-1}}%
{1}%
{6.045998\times10^{-1}}%
{16/27}%
{2}%
{27/64}%
{8.306815\times10^{-1}}%
{9.540383\times10^{-1}}%
{7.827314\times10^{-1}}%
{1}%
{4.105301}%
{1.895795\times10^{-1}}%
{1.214249}%
{0.980969}%