i = I;

E1tilde[x_, y_, u_, v_] := y^2 - x^3 - x^2 - x + 3/4;
alpha = 1;
beta = 3/2;
mu = 6;
b1[x_, y_, u_, v_] = 
        2*beta*(y - beta) - mu*(x - alpha);
b2[x_, y_, u_, v_] = -2*beta*(y + beta) - 
          mu*(x - alpha);
omega[1, x_, y_, u_, v_] = 
        b1[x, y, u, v]*b2[x, y, u, v] + 4*beta^2*b2[x, y, u, v]*u^4 + 
          4*beta^2*b1[x, y, u, v]*v^4;
omega[2, x_, y_, u_, v_] = -4*beta^2*(x - alpha)^2*
          u v*(1 + u + v);
omega[3, x_, y_, u_, v_] = 
        b1[x, y, u, v]*b2[x, y, u, v]*v + 4*beta^2*b2[x, y, u, v]*u^3 + 
          4*beta^2*b1[x, y, u, v]*v^3  u;
omega[4, x_, y_, u_, v_] = 
        b1[x, y, u, v]*b2[x, y, u, v]*u + 4*beta^2*b2[x, y, u, v]*u^3  v + 
          4*beta^2*b1[x, y, u, v]*v^3;
omega[5, x_, y_, u_, v_] = 
        4*beta^2*(x - alpha)*(4*beta^2*u^2  v^2 + b1[x, y, u, v]*v^2 + 
              b2[x, y, u, v]*u^2);

Given a point in \tilde P, one can ask whether a divisor given by an equation  \(TraditionalForm\`\(\(\(\(a\_1\) ω\_1\)\(+\)\) ... \) + \(a\_5\) ω\_5\)  (1) passes through the point and (2) is singular at the point.  This amounts to four linear equations in the variables (Subscript[a, 1]),  ...  ,  (Subscript[a, 5]).  The function below assigns to a point with coordinates (x, y, u, v)the corresponding 4-by-5 matrix of this linear system:

derivmat[xVal_, yVal_, uVal_, vVal_] := 
      Limit[Limit[
          Limit[Limit[
Transpose[
                Table[{omega[i, x, y, u, 
                      v], (∂_y) E1tilde[x, y, u, v]*(∂_x) 
                          omega[i, x, y, u, v] - (∂_x) 
                          E1tilde[x, y, u, v]*(∂_y) 
                          omega[i, x, y, u, v], (∂_u) 
                      omega[i, x, y, u, v], (∂_v) 
                      omega[i, x, y, u, v]}, {i, 1, 5}]]
, 
              x -> xVal]
, 
            y -> yVal]
, 
          u -> uVal]
, v -> vVal]

The kernel of this matrix has dimension at least 1, and has dimension exactly 1 if and only if one of its 4-by-4 minors does not vanish.  The function below computes the minor with the ith colum deleted:

derivminor[i_, x_, y_, u_, 
        v_] := Minors[derivmat[x, y, u, v], 4][[1, 6 - i]]

This linear algebra can be translated into geometric language as follows.  One potentially has a rational map from \tilde P to P^4 that sends a point to the unique divisor in our chosen linear system that passes through and is singular at that point.  This rational map sends the point to the (projectivized) kernel of the associated matrix above.  The caveat "potentially" comes in because one must show it is defined at least one point, i.e., one must show that for at least one point of \tilde P the matrix above has a one-dimensional kernel.  We will show that it is in fact defined at 126 points of \tilde P, namely those coming from singular divisors in the pencil \tilde (Subscript[J, 1]).

With this in mind, here is a condensed version of the data in the file "NUMERICAL_DATA.pdf".  It contains the coordinates (x, y, u, v, t, a) for all points of T that are singularities of some fiber of \tilde (Subscript[J, 1]).

pencildata = 
        ToExpression["{{-0.727117404359727-i*1.062822946727292,-0.\
0474342875874572+i*0.0254504866453305,0.42196879333238+i*0.140527099177768,-0.\
518616520553573-i*0.170138267119806,-0.419964985922571+i*0.25843548489165,0.\
641034515442334+i*0.224344980006387},{-1.025692664161746+i*0.613064294841613,\
0.47453805968051+i*1.116800621887435,-1.227210517261125+i*0.00549345588578142,\
2.133250760378274-i*0.710430595607088,-0.452236363351381-i*0.136866747905453,\
0.641034515442334+i*0.224344980006387},{-1.027651960961238+i*0.\
534552137918721,-0.415117163088272-i*1.176426920557871,-1.740848922069183+i*0.\
571106023776653,-0.81483979186379-i*0.00364752940723909,-0.461131984059381-i*\
0.121568736986197,0.641034515442334+i*0.224344980006387},{-0.727117404359727+\
i*1.062822946727292,-0.0474342875874572-i*0.0254504866453305,0.42196879333238-\
i*0.140527099177768,-0.518616520553573+i*0.170138267119806,-0.419964985922571-\
i*0.25843548489165,0.641034515442334-i*0.224344980006387},{-1.025692664161746-\
i*0.613064294841613,0.47453805968051-i*1.116800621887435,-1.227210517261125-i*\
0.00549345588578142,2.133250760378274+i*0.710430595607088,-0.452236363351381+\
i*0.136866747905453,0.641034515442334-i*0.224344980006387},{-1.\
027651960961238-i*0.534552137918721,-0.415117163088272+i*1.176426920557871,-1.\
740848922069183-i*0.571106023776653,-0.81483979186379+i*0.00364752940723909,-\
0.461131984059381+i*0.121568736986197,0.641034515442334-i*0.224344980006387},{\
2.385369974863955+i*0.441361180987026,-4.53056697526151-i*1.103065647246238,0.\
681138931444903-i*0.464692280165189,-0.106206683114771+i*0.449148340779966,0.\
655315720113476-i*0.208775219180746,0.568893317488144-i*0.943829416456446},{\
12.818092929007386-i*4.926176851849022,45.284541136532436-i*26.\
938931921581205,-0.498589939919732-i*2.108538160470867,-1.319430424510625-i*1.\
2045165334852,0.0720903173400666+i*0.0300496581518177,0.568893317488144-i*0.\
943829416456446},{0.518360120309197-i*0.0417720741069623,0.43632123494839-i*0.\
135997575659869,-0.413386902064433+i*0.377383565675711,1.001838200025711+i*0.\
683482408704743,-2.060739370786876+i*0.178725561028928,0.568893317488144-i*0.\
943829416456446},{2.385369974863955-i*0.441361180987026,-4.53056697526151+i*1.\
103065647246238,0.681138931444903+i*0.464692280165189,-0.106206683114771-i*0.\
449148340779966,0.655315720113476+i*0.208775219180746,0.568893317488144+i*0.\
943829416456446},{12.818092929007386+i*4.926176851849022,45.284541136532436+i*\
26.938931921581205,-0.498589939919732+i*2.108538160470867,-1.319430424510625+\
i*1.2045165334852,0.0720903173400666-i*0.0300496581518177,0.568893317488144+i*\
0.943829416456446},{0.518360120309197+i*0.0417720741069623,0.43632123494839+i*\
0.135997575659869,-0.413386902064433-i*0.377383565675711,1.001838200025711-i*\
0.683482408704743,-2.060739370786876-i*0.178725561028928,0.568893317488144+i*\
0.943829416456446},{0.989762347378112-i*1.424415068693428,-1.049545658290926+\
i*2.639331673117822,0.427459760121224-i*0.832149543434724,-0.390365150781157+\
i*0.465652022914856,-0.00504550727611333+i*0.7020062956554,0.360525673667314-\
i*1.34970228648487},{-2.85551556065106+i*8.0597432043369,-23.126643438289809+\
i*7.877701340007325,-1.05727907243868-i*1.261188755915909,-1.501441906106259+\
i*0.340706854341537,-0.0482999261633129-i*0.100968338874762,0.360525673667314-\
i*1.34970228648487},{0.359883035372407+i*0.300576741820982,0.403565080105162+\
i*0.751494620144886,-0.633410492748245-i*0.143733364316997,0.48841675531923+i*\
0.950816469436807,-1.279987899893907-i*0.601037956780638,0.360525673667314-i*\
1.34970228648487},{0.989762347378112+i*1.424415068693428,-1.049545658290926-i*\
2.639331673117822,0.427459760121224+i*0.832149543434724,-0.390365150781157-i*\
0.465652022914856,-0.00504550727611333-i*0.7020062956554,0.360525673667314+i*\
1.34970228648487},{-2.85551556065106-i*8.0597432043369,-23.126643438289809-i*\
7.877701340007325,-1.05727907243868+i*1.261188755915909,-1.501441906106259-i*\
0.340706854341537,-0.0482999261633129+i*0.100968338874762,0.360525673667314+i*\
1.34970228648487},{0.359883035372407-i*0.300576741820982,0.403565080105162-i*\
0.751494620144886,-0.633410492748245+i*0.143733364316997,0.48841675531923-i*0.\
950816469436807,-1.279987899893907+i*0.601037956780638,0.360525673667314+i*1.\
34970228648487},{1.359402124563056+i*1.642352759083765,-1.603812846531419-i*3.\
361585959529432,0.702323701813534-i*0.411935381764477,0.256419507489642+i*0.\
215166111527453,0.127154813986308-i*0.581056831077682,0.315520084051802-i*0.\
53189051026745},{0.259894006648671-i*8.104109962589794,-14.389381467149381-i*\
18.009518780157693,2.288490970420659-i*1.920312959773512,0.816216597721111-i*\
2.291391708113429,-0.0111757372568355+i*0.122373557917417,0.315520084051802-i*\
0.53189051026745},{0.372835513187716-i*0.198487129223209,0.339566831551303-i*\
0.620564587383241,0.137951662039964+i*0.387276239421496,1.05939271450643+i*0.\
621367812536925,-1.449312410062806+i*0.458683273160266,0.315520084051802-i*0.\
53189051026745},{1.359402124563056-i*1.642352759083765,-1.603812846531419+i*3.\
361585959529432,0.702323701813534+i*0.411935381764477,0.256419507489642-i*0.\
215166111527453,0.127154813986308+i*0.581056831077682,0.315520084051802+i*0.\
53189051026745},{0.259894006648671+i*8.104109962589794,-14.389381467149381+i*\
18.009518780157693,2.288490970420659+i*1.920312959773512,0.816216597721111+i*\
2.291391708113429,-0.0111757372568355-i*0.122373557917417,0.315520084051802+i*\
0.53189051026745},{0.372835513187716+i*0.198487129223209,0.339566831551303+i*\
0.620564587383241,0.137951662039964-i*0.387276239421496,1.05939271450643-i*0.\
621367812536925,-1.449312410062806-i*0.458683273160266,0.315520084051802+i*0.\
53189051026745},{-1.652634601548776+i*0.671414019057143,0.992130829617419+i*1.\
839897886562361,0.293950587472331-i*1.2106200409311,0.281701592759247-i*0.\
614005054534482,-0.354286091446543-i*0.0896741105673902,0.135833211086297-i*0.\
722880985759328},{-0.596002883843082-i*1.089671850482931,-0.288826958730771-i*\
0.591816541150249,0.617282037879421+i*1.345446036065653,1.810274352923163-i*0.\
351799353249802,-0.427354638504571+i*0.291776615484816,0.135833211086297-i*0.\
722880985759328},{-0.598135517137693+i*0.585447746137834,-0.159819041386233-i*\
0.97858165429602,0.532299616169901+i*0.103444353835817,0.189400196010017+i*0.\
780034750117955,-0.551692603382219-i*0.202102504917426,0.135833211086297-i*0.\
722880985759328},{-1.652634601548776-i*0.671414019057143,0.992130829617419-i*\
1.839897886562361,0.293950587472331+i*1.2106200409311,0.281701592759247+i*0.\
614005054534482,-0.354286091446543+i*0.0896741105673902,0.135833211086297+i*0.\
722880985759328},{-0.596002883843082+i*1.089671850482931,-0.288826958730771+i*\
0.591816541150249,0.617282037879421-i*1.345446036065653,1.810274352923163+i*0.\
351799353249802,-0.427354638504571-i*0.291776615484816,0.135833211086297+i*0.\
722880985759328},{-0.598135517137693-i*0.585447746137834,-0.159819041386233+i*\
0.97858165429602,0.532299616169901-i*0.103444353835817,0.189400196010017-i*0.\
780034750117955,-0.551692603382219+i*0.202102504917426,0.135833211086297+i*0.\
722880985759328},{-0.957793686752595-i*1.110169487449516,-0.885064881874663+i*\
0.378836427348196,-0.0468334464280645-i*0.601326734347402,-0.238371981061922+\
i*0.879170630450911,-0.386500801510419+i*0.219165788313154,0.133743444476267-\
i*1.2703608592736},{-0.939492918054843+i*1.117640434849119,0.864984774916842+\
i*0.335837384850484,-1.422548687909944-i*0.507203191555197,-0.128738572263579+\
i*1.65296281072783,-0.387066165247532-i*0.223048402608501,0.133743444476267-i*\
1.2703608592736},{-0.786373658858756-i*0.0123905263318126,0.00670397353389515-\
i*1.18495018108538,-0.287277465044611-i*1.059545291071777,-0.623678716004202+\
i*0.222369777534394,-0.559766366575382+i*0.00388261429534645,0.\
133743444476267-i*1.2703608592736},{-0.957793686752595+i*1.110169487449516,-0.\
885064881874663-i*0.378836427348196,-0.0468334464280645+i*0.601326734347402,-\
0.238371981061922-i*0.879170630450911,-0.386500801510419-i*0.219165788313154,\
0.133743444476267+i*1.2703608592736},{-0.939492918054843-i*1.117640434849119,\
0.864984774916842-i*0.335837384850484,-1.422548687909944+i*0.507203191555197,-\
0.128738572263579-i*1.65296281072783,-0.387066165247532+i*0.223048402608501,0.\
133743444476267+i*1.2703608592736},{-0.786373658858756+i*0.0123905263318126,0.\
00670397353389515+i*1.18495018108538,-0.287277465044611+i*1.059545291071777,-\
0.623678716004202-i*0.222369777534394,-0.559766366575382-i*0.\
00388261429534645,0.133743444476267+i*1.2703608592736},{-0.719567633338283-i*\
1.061187771245258,-0.0432437934068117-i*0.146281075779216,0.731324401108206+i*\
0.0547166694724529,1.134008581770614+i*0.501763092788966,-0.421149649021088+i*\
0.259901878088846,0.0455229307296027-i*0.0923174095065175},{-1.\
143032996732431+i*0.554500269613977,0.510800194641771+i*1.2625171445335,1.\
593038836769857+i*0.566913004232311,1.35977048823019-i*0.101736127291996,-0.\
437348230576291-i*0.113161912177514,0.0455229307296027-i*0.0923174095065175},{\
-0.922400425792338+i*0.594085742185468,-0.379158058202802-i*1.061331964365267,\
0.737450644486099-i*0.326298691301623,0.557169628566901-i*0.198279351831958,-\
0.474835453735954-i*0.146739965911332,0.0455229307296027-i*0.0923174095065175}\
,{-0.719567633338283+i*1.061187771245258,-0.0432437934068117+i*0.\
146281075779216,0.731324401108206-i*0.0547166694724529,1.134008581770614-i*0.\
501763092788966,-0.421149649021088-i*0.259901878088846,0.0455229307296027+i*0.\
0923174095065175},{-1.143032996732431-i*0.554500269613977,0.510800194641771-i*\
1.2625171445335,1.593038836769857-i*0.566913004232311,1.35977048823019+i*0.\
101736127291996,-0.437348230576291+i*0.113161912177514,0.0455229307296027+i*0.\
0923174095065175},{-0.922400425792338-i*0.594085742185468,-0.379158058202802+\
i*1.061331964365267,0.737450644486099+i*0.326298691301623,0.557169628566901+i*\
0.198279351831958,-0.474835453735954+i*0.146739965911332,0.0455229307296027+i*\
0.0923174095065175},{-0.92233517313235-i*1.059372921187273,0.742371506856815-\
i*0.417518818348412,-2.0600109821272-i*0.516466721514277,0.881722741824418+i*\
0.471767958375334,-0.399019805332964+i*0.219894419399492,-i*0.157582936880624}\
,{-0.92233517313235+i*1.059372921187273,-0.742371506856815-i*0.\
417518818348412,0.881722741824418-i*0.471767958375334,-2.0600109821272+i*0.\
516466721514277,-0.399019805332964-i*0.219894419399492,-i*0.157582936880624},{\
-0.868133246578968,i*1.23237622685211,-0.456726385963565-i*0.114506175566156,-\
0.456726385963565+i*0.114506175566156,-0.535293722667404,-i*0.157582936880624}\
,{-0.92233517313235-i*1.059372921187273,-0.742371506856815+i*0.\
417518818348412,0.881722741824418+i*0.471767958375334,-2.0600109821272-i*0.\
516466721514277,-0.399019805332964+i*0.219894419399492,i*0.157582936880624},{-\
0.92233517313235+i*1.059372921187273,0.742371506856815+i*0.417518818348412,-2.\
0600109821272+i*0.516466721514277,0.881722741824418-i*0.471767958375334,-0.\
399019805332964-i*0.219894419399492,i*0.157582936880624},{-0.868133246578968,-\
i*1.23237622685211,-0.456726385963565+i*0.114506175566156,-0.456726385963565-\
i*0.114506175566156,-0.535293722667404,i*0.157582936880624},{2.8863420626876-\
i*3.287183262855727,3.7590337553259-i*9.164513733352289,-0.593758577165281-i*\
0.804643245197933,-0.622190101741968+i*1.2323732501286,0.13132556561463+i*0.\
22885096495087,-i*1.857073682646946},{2.8863420626876+i*3.287183262855727,-3.\
7590337553259-i*9.164513733352289,-0.622190101741968-i*1.2323732501286,-0.\
593758577165281+i*0.804643245197933,0.13132556561463-i*0.22885096495087,-i*1.\
857073682646946},{0.373427484913478,-i*0.430175174067733,-0.326460853119771-i*\
0.646621702069147,-0.326460853119771+i*0.646621702069147,-1.595984464562593,-\
i*1.857073682646946},{2.8863420626876-i*3.287183262855727,-3.7590337553259+i*\
9.164513733352289,-0.622190101741968+i*1.2323732501286,-0.593758577165281-i*0.\
804643245197933,0.13132556561463+i*0.22885096495087,i*1.857073682646946},{2.\
8863420626876+i*3.287183262855727,3.7590337553259+i*9.164513733352289,-0.\
593758577165281+i*0.804643245197933,-0.622190101741968-i*1.2323732501286,0.\
13132556561463-i*0.22885096495087,i*1.857073682646946},{0.373427484913478,i*0.\
430175174067733,-0.326460853119771+i*0.646621702069147,-0.326460853119771-i*0.\
646621702069147,-1.595984464562593,i*1.857073682646946},{-0.913197500713585-i*\
1.043885687513148,-0.702848408217753+i*0.434944712780275,-0.808737294403166-i*\
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0.710430595607088,-1.227210517261125+i*0.00549345588578142,-0.452236363351381-\
i*0.136866747905453,-0.641034515442334-i*0.224344980006387},{-1.\
027651960961238+i*0.534552137918721,0.415117163088272+i*1.176426920557871,-0.\
81483979186379-i*0.00364752940723909,-1.740848922069183+i*0.571106023776653,-\
0.461131984059381-i*0.121568736986197,-0.641034515442334-i*0.224344980006387},\
{-0.727117404359727+i*1.062822946727292,0.0474342875874572+i*0.\
0254504866453305,-0.518616520553573+i*0.170138267119806,0.42196879333238-i*0.\
140527099177768,-0.419964985922571-i*0.25843548489165,-0.641034515442334+i*0.\
224344980006387},{-1.025692664161746-i*0.613064294841613,-0.47453805968051+i*\
1.116800621887435,2.133250760378274+i*0.710430595607088,-1.227210517261125-i*\
0.00549345588578142,-0.452236363351381+i*0.136866747905453,-0.641034515442334+\
i*0.224344980006387},{-1.027651960961238-i*0.534552137918721,0.\
415117163088272-i*1.176426920557871,-0.81483979186379+i*0.00364752940723909,-\
1.740848922069183-i*0.571106023776653,-0.461131984059381+i*0.121568736986197,-\
0.641034515442334+i*0.224344980006387}}"];

The following function evaluates (in chunks of 3) the aforementioned rational map on \tilde P at the 126 points above in T.  It is normalized so that the third column (Subscript[a, 3]) is always 1.  Note that in all cases, the first column (Subscript[a, 1]) is nonzero, meaning that the rational map is defined at each of the 126 points.

As a check, also note that, as expected with this normalization, the entry in the first column corresponds to the a-coordinate of the full solution vector (x, y, u, v, t, a).  This is because, the rational map being defined at (x, y, u, v), there is only one singular divisor in the linear system containing that point as a singularity, and so necessarily this must be the divisor \(TraditionalForm\`a\ ω\_1 + \((ω\_3 - ω\_4)\)\) in the pencil \tilde J_1.

Do[
Print[
MatrixForm[
Table[
Table[
derivminor[
p, pencildata[[q, 1]], 
                  pencildata[[q, 2]], pencildata[[q, 3]], 
                  pencildata[[q, 
                      4]]
], 
{p, 1, 
                  5}
]
/
              derivminor[3, pencildata[[q, 1]], pencildata[[q, 2]], 
                pencildata[[q, 3]], 
                pencildata[[q, 4]]], 
{q, 3  k + 1, 
              3  k + 3}]
]
], 
{k, 0, Length[pencildata]/3 - 1}]