A few examples of heights computations for some particular curves.

Examples from Silverman's paper "Computing heights on elliptic curves" Mathematics of Computation 183 (1988) 339--358.

//First curve
> PQ<x>:=PolynomialRing(Rationals());
> K1<t>:=NumberField(x^2+2);
> E1:=EllipticCurve([0,-1,1,0,0]);
> P1:=E1(K1)![2+t,1+2*t];
> time Hauteur(P1,50)/2;
0.45754773287523276736211210741423654346576029814695111199692143853256079561821
Time: 0.100

// Second curve
> PQ<x>:=PolynomialRing(Rationals());
> K2<i>:=NumberField(x^2+1);
> E2:=EllipticCurve([0,0,4,6*i,0]);
> P2:=E2(K2)![0,0];
> time Hauteur(P2,50)/2;
0.3368982000223863870359242354034550243558328030851613945244296457283
Time: 0.130

> Q2:=E2(K2)![-9/4,(-32+27*i)/8];
> time Hauteur(Q2,50)/2;
1.347592800089545548143696941613820097423331212340645578101329704314
Time: 0.090

//Third curve
> E3:=EllipticCurve([21,26,494,0,0]);
> P3:=E3![0,0];
> time Hauteur(P3,50)/2;
0.01049206065917646825752714903456547617902004085196629223786536421697
Time: 0.080

In particular, for the curves from the preprint Group of Weierstrass points of a plane quartic with eight hyperflexes or more

Computation of the heights of some images of the Weierstrass points on the elliptic factors of the Jacobian of a specialisation of the curve Z_1 of section 4 (corresponding to t=2/5).

> PQ<x> := PolynomialRing(RationalField());
> K<i>:= NumberField(x^2+1);
> PK<x> := PolynomialRing(K);
> L<sq35>:=NumberField(x^2-35);
> PL<x> := PolynomialRing(L);
> M<sq65>:= NumberField(x^2-65);
> AM:=AbsoluteField(M);   
> OAM:=MaximalOrder(AM); 

> E1:=EllipticCurve([0,0,0,-247,-1386]);
> Q11:=E1(AM)![10/9*i*sq35-73/9,40/27*i*sq35-940/27,1]; 
> time Hauteur(Q11,50); 
1.084825643005921430632180416169393752995539457428612689694061041344
Time: 17.610
> Q13:=E1(AM)![2/5*i*sq35*sq65-1,8/5*i*sq35*sq65-76,1];
> time Hauteur(Q13,50);                                   
0.9677433910169370903140314347834855385085164933041984466915334852674
Time: 0.510
> time Hauteur(Q11+Q13,50);
2.052569034022858520946211850952879291504055950732811136203434569790
Time: 0.490

> E2:=EllipticCurve([AM|0,0,0,253,-6286]);                  
> Q21:=E2(AM)![(1/45*(19*i-9)*sq35+1/9*(-9*i+7))*sq65+1/9*(-27*i+1)*sq35+163/9*i,
      (1/135*(-77*i+301)*sq35+1/27*(359*i-359))*sq65+1/27*(149*i-581)*sq35+1/27*(-977*i+3721),1];
> time Hauteur(Q21,50);       
1.026284517011429260473105925476439645752027975366405568010936380197
Time: 28.350
> Q22:=E2(AM)![(1/45*(19*i-9)*sq35+1/9*(9*i-7))*sq65+1/9*(27*i-1)*sq35+163/9*i,
      (1/135*(-77*i+301)*sq35+1/27*(-359*i+359))*sq65+1/27*(-149*i+581)*sq35+1/27*(-977*i+3721),1];
> time Hauteur(Q22,50);
1.026284517011429260473105925476439645752027975366405568010936380197
Time: 0.430
> time Hauteur(Q21+Q22,50);
1.935486782033874180628062869566971077017032986608396893019022295318
Time: 0.420

> E3:=EllipticCurve([0,-1,0,-220,832]);
> Q31:=E3(AM)![10/9*i*sq65 - 4/9, -100/27*i*sq65 + 1210/27,1];
> time Hauteur(Q31,50);
1.035403179541875057469231459744335970454003980860049420448279459854
Time: 16.030

Computation of the heights of the images of some Weierstrass points on the elliptic factors of the Jacobian of the curve C of section 5.

> PQ<x> := PolynomialRing(RationalField());
> MM<m>:=NumberField(x^16-280*x^12+45584*x^8+878080*x^4+9834496);
> a:=1/256*m^4;
> b:=-1/4*m;
> t:=1/1056048*(5*m^12-1498*m^8+239680*m^4+2156784);

> sub := Subfields(MM); 
> exists(Ft){F[1] : F in sub | Degree(F[1]) eq 2 and t in F[1]};
true
> Ft;
Number Field with defining polynomial x^2 - 784*x + 9834496 over the Rational 
Field
> t:=Ft!t; 
> y0:=b;
> x0:=-((-160/49*t + 32/7)*y0^7 + (20/7*t + 20/7)*y0^3);
> PP<X,Y,Z>:=ProjectivePlane(Ft);
> g:=-4*X^4-24*t*X^2*Y^2+(12*t-12)*X*Y*Z^2-4*Y^4+(-3/2*t +7/2)*Z^4;
> CC:=Curve(PP,g);
> Q1:=CC(MM)![x0,y0,1];
> F1:=EllipticCurve([0,0,0,-19,-30]);

> phi1:=map<CC->F1|[-2*X^3 + t*X*Y^2 ,6*t*X*Y^2 + 1/4*(-7*t+7)*Y*Z^2,X^3]>;
> Q11:=F1(MM)!phi1(Q1);

> time Hauteur(Q11,50);
0.5345542883986127802962826814094425633983246410493268360883984462364
Time: 829.860

> PQ<x> := PolynomialRing(RationalField());
> MM2<m>:=NumberField(x^32 + 6*x^30 + 29*x^28 + 112*x^26 + 126*x^24 -
         140*x^22 - 1078*x^20 - 5104*x^18 - 6719*x^16 + 12550*x^14 + 47845*x^12
         + 23856*x^10 - 7952*x^8 + 47040*x^6 + 72052*x^4 - 31088*x^2 + 8464);
>t:=1/884002144864506112000*(-110489391148423029*m^30 -
    709865486686283623*m^28 - 3442880329575341104*m^26 -
    13529093027734989072*m^24 - 18157680027082689686*m^22 +
    13383343646565276894*m^20 + 127419430729434221676*m^18 +
    610214978372991570172*m^16 + 946072309125992828983*m^14 -
    1269165534454111154427*m^12 -  5991996516530836766492*m^10 -
    4436434720295736066876*m^8 + 855055425717775188052*m^6 -
    4304234778661262922748*m^4 - 8390220256414811432096*m^2 +
    1566900478903887719376); 
> a:=1/16859688906855860568064000*(428004701513779943211*m^30 + 
    2760842205527733650257*m^28 + 13636449543533389745136*m^26 + 
    53872444306318549919248*m^24 + 77284460024951209254474*m^22 - 
    29324756884053938421746*m^20 - 483529388574247894536884*m^18 - 
    2401425288331226994126148*m^16 - 3931631370226263032095497*m^14 + 
    3780428214180962647125693*m^12 + 22630020469003742224845028*m^10 + 
    20333869025742378872051684*m^8 + 3575291492677679157910932*m^6 + 
    19384772015174468036764132*m^4 + 39855949120633597485149664*m^2 + 
    2547239012215605974612816);
> b:=1/15535149597110997022185472000*(1177032126640176279623029*m^31 + 
    7385719391222410024829593*m^29 + 36054629668717252913803864*m^27 + 
    141130261257694093603045752*m^25 + 183952519254695463465536926*m^23 - 
    126228344701100154402538874*m^21 - 1317172372941691686489614676*m^19 - 
    6367655919287331354454445852*m^17 - 9538317077319404250638674223*m^15 + 
    12718715185330193206614283197*m^13 + 60505253244348561493281135772*m^11 + 
    44169664036301828389205283316*m^9 - 2150688543258812231750712132*m^7 + 
    49626031233263970278250848708*m^5 + 93857358777575806785007472896*m^3 - 
    11326776777164946782471717296*m);
> i:=1/31954659346414592*(2166546548543*m^31 + 13327888802868*m^29 + 
    65118435819808*m^27 + 254118422374852*m^25 + 318702595665006*m^23 - 
    226235355685964*m^21 - 2343656733134808*m^19 - 11467130324786436*m^17 - 
    16513810398887161*m^15 + 23261669256934584*m^13 + 105494466368955944*m^11 + 
    71763232136519504*m^9 + 1979033061344252*m^7 + 106199169112690192*m^5 + 
    181210047049573184*m^3 - 8977007357238624*m);
> i^2;
-1
> t^2;
-7
> (a^2 + (15/64*t - 35/64)*a + 147/8192*t + 49/8192);
0
> (b^4 - a);
0

> sub := Subfields(MM2);
> exists(Ft){F[1] : F in sub | Degree(F[1]) eq 2 and t in F[1]};
true
> Ft;
Number Field with defining polynomial x^2 - 177*x + 8464 over the
    Rational Field
> t0:=Ft!t;
> E:=EllipticCurve([0,0,0,5,-2*t0]);
> E;
Elliptic Curve defined by y^2 = x^3 + 5*x + 1/19*(-4*Ft.1 + 354) over Ft

> y0:=b;
> x0:=-((-160/49*t0 + 32/7)*y0^7 + (20/7*t0 + 20/7)*y0^3);
> PP<X,Y,Z>:=ProjectivePlane(Ft);
> g:=-4*X^4-24*t0*X^2*Y^2+(12*t0-12)*X*Y*Z^2-4*Y^4+(-3/2*t0 +7/2)*Z^4;
> CC:=Curve(PP,g);
> Q1:=CC(MM2)![x0,y0,1];Q2:=CC(MM2)![-i*x0,i*y0,1];

> phi2:=map<CC->E|[(t0+2)*(X+Y)^2+(t0+1)*(X+Y)*Z-Z^2,(-t0+1)*(X-Y)^2+(3*t0+9)*(-2*X*Y+1/2*Z^2),(X+Y+Z)^2]>;   
> Q22:=E(MM2)!phi2(Q2);
> Q21:=E(MM2)!phi2(Q1);

> time Hauteur(Q21,50);
0.2605661397687820081879578069601416967921784819609867582779109040081
Time: 2230.390
> time Hauteur(Q22,50);    
0.2605661397687820081879578069601416967921784819609867582779109040067
Time: 361.900
> time Hauteur(Q22+Q21,50);
0.5211322795375640163759156139202833935843569639219735160168654100055
Time: 210.420

Computation of the height over the rationals of a point with large coordinates

> E:=EllipticCurve(CremonaDatabase(),5077,1,1);
> P:=E![
       393905805863577276581630023578327436065312418604341874732254673308507170874781\
       6746184264880266146098367135082065582096608052286390718325291335860624786354804\
       1611940472759194702426134371231378466576390175374916460490052466749845132233879\
       6639216293219649800641280707120491328735293279842980839006866136817326720265664\
       3655916808926940342071292726845418636332631759564114992663129228618347124501900\
       432412516084928193522925388801150/487317528244829968342728065703777780077138546\
       1021265537663597618796926718293320528718991731186785058436842898587062441846191\
       4031758469928020096986811321576320171668249735933274213246374231934603624381553\
       8839904766274447761012373829417083437399779522614911200284696311842821597107963\
       2811904103020136658256416874149599469826283311715275365215028753401063262276395\
       126164423438785358322511312019727303314905134740800178346904025649 ,
       -167633155188263744882209218775727122426710674379681821188899484050002954095326\
       6368821642382773404489272948739377436832606030643944044110809644719852626918450\
       1225395949588394280063928926914092874109741444355132714070651814975036910310954\
       3117291163208151763442781987259335924593199942274199993397388701546188973582281\
       4840812161472161058277852829198852703674846635930761721974582034040628202881439\
       7378785605394086400141977127971026789224858333340027512687184219726505321177438\
       8709023119615404459960049982561278554309112464032913599285192167986089575390179\
       8507555575424743019422841535401322131809781263348314991969234449912971780571212\
       5609421244/10757665820476340492037069907661107416398592258062103891822933189023\
       1669058113172060066228972833264611429975818821269938264472297113241602013247363\
       8740734512427010294051385300574442708263629465613375040155140836714388754840876\
       3592578552272329053572632440192064136236056573007546168134939394967910395414551\
       7758350926225331603730559479838717398985730121517023261491505468050377030154904\
       8263011335110293199821928047781821728196446136595477233818344314252003523181570\
       6040053576419480251720615211093261187269804922712867803158409128075657265857519\
       3700548058008301265170565593621499071151585725979411458907080819980177162015771\
       32992599665923949143 , 1]; 

> time Hauteur(P,50);
983.22978602196008655792309027796665018020176888007799999996672
Time: 0.060

> time Hauteur(2*P,50)/4;
983.22978602196008655792309027796665018020176888007749999998005
Time: 0.230

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