ell

default(parisize,"20M");
default(realprecision,38);
/* Test generic */

F={
  [ellwp, ellzeta, ellsigma,
  ellformalw, ellformalpoint, ellformaldifferential, ellformallog, ellformalexp,
  E->ellisoncurve(E,[0,0]), E->ellordinate(E,0),
  E->elldivpol(E,5), E->ellxn(E,3), E->ellmul(E,[0,0],0), E->ellneg(E,[0,0])
  ];
}

L={
  [1,I,'z,Mod('z,'z^2+5),
   ellinit([0,0,0,'z,0],nfinit('z^2+5)),
   Mod(4,1009),ffgen([1009,5],'z)];
}
L=apply(x->if(type(x)=="t_VEC",x,ellinit([x,0])),L);
for(j=1,#F,for(i=1,#L, print(F[j],":",L[i][4],":",F[j](L[i]))));

ellap(ellinit([1,0,1,4,-6]), 2)
ellap(ellinit([0,17]), 2)
ellap(ellinit([0, 0, 1, -1, 0]),2486246173)

testgroup(v,p)=
{ my(E,F,G,g);
  E=ellinit(v); if(!E || E.disc%p==0, next);
  G=ellgroup(E,p,1); if (G.no == 1, return);
  g=G.gen; if (!ellisoncurve(E,g), error(E, G));
  F=vector(#g,i, ellorder(E,g[i],G.no));
  if ((p <= 3 && F!=G.cyc) || (p > 3 && F[1] != G.cyc[1]),
    print("error:",v,":",F,":",G));
}
{
  for(x=10,63,
    my(p,E,G,g);
    p=nextprime(2^x);E=ellinit([0,0,1,2,3]);
    G = ellgroup(E,p,1); if (!ellisoncurve(E,G.gen), error(E));
    if(ellorder(E,G.gen[1])!=G.cyc[1], error(E));
    print(p,":",ellgroup(E,p)));
  for(p=2,3,
    forvec(v=vector(5,i,[0,p-1]), testgroup(v,p)));
  forvec(v=vector(2,i,[0,4]), testgroup(v,5));
}
setrand(1)
a=ffgen(2^8,'a); E=ellinit([a,1,0,0,1]);
P=[a^3,ellordinate(E,a^3)[1]]; Q=ellmul(E,P,113);
e=elllog(E,P,Q,242)
ellmul(E,Q,e) == P
ellpow(E,Q,e) == P

p=655637;
E=ellinit([123,47], p);
X=1;until(Y!=[],X++;Y=ellordinate(E,X));
P=[X,Y[1]]; Q=ellmul(E,P,113);
o=ellorder(E,P, p+1-ellap(E,p))
e=elllog(E,P,Q,o)
ellmul(E,Q,e) == P

p=1073741827;
E=ellinit([1,3], p);
G=[Mod(1050932506,p),Mod(12325986,p)];
P=ellmul(E,G,1023);
elllog(E,P,G)

ellorder(ellinit([0,2],1),[-1,-1]*Mod(1,997))

ellinit([a1,a2,a3,a4,a6]*Mod(1,5));
ellinit([a1,a2,a3,a4,a6]);
ellinit([0]).j
ellinit([1728]).j
ellinit([j]).j
ellinit([Mod(0,2)]).j
ellinit([Mod(0,5)]).j
ellinit([Mod(3,5)]).j
ellinit(['j*Mod(1,2)]).j
ellinit([Mod(0,3)]).j
ellinit([3], 3).j
ellinit([1728], ffgen(3^5,'a)).j
ellinit(['j*Mod(1,3)]).j

elltwist([0,a2,0,a4,a6],x^2-D/4)[1..5]
lift(elltwist([a1,a2,a3,a4,a6]*Mod(1,2),x^2+x+T)[1..5])
elltwist([a1,a2,a3,a4,a6],1)[1..5]
elltwist([a1,a2,a3,a4,a6],-4)[1..5]
elltwist([a1,a2,a3,a4,a6],5)[1..5]
elltwist([a4,a6],5)[1..5]

e = ellinit([4,0]);
ellminimaltwist(e)
ellminimaltwist(e,1)

e = ellinit([-882216989/131072]);
e = ellintegralmodel(e,&v)
v
ellminimaltwist(e)
ellminimaltwist(e,1)

twisttest(c4,c6, T)=
{ my(e = ellinit([c4/-48, c6/-864]));
  vector(#T, i,
    my (et = ellinit(elltwist(e, T[i])));
    my (d = ellminimaltwist(et));
    et = ellinit(elltwist(et,d));
    [d, ellminimalmodel(et).disc]);
}
T3 = [1,-3];
twisttest(-4*3,8*3^3, T3)
twisttest(0,8*3^3, T3)
twisttest(-4*3,0, T3)
twisttest(16,-8*19, T3)
twisttest(-4*3^4, 8*3^6, T3)
twisttest(4*3^6, 16*3^9, T3)
T2 = [1,-4,8,-8];
a=2;b=3; twisttest(3*2^a, 9*2^b, T2)
a=4;b=5; twisttest(3*2^a, 9*2^b, T2)
b=5; twisttest(0, 9*2^b, T2)
a=4; twisttest(3*2^a, 0, T2)
twisttest(1, -161, T2)
d=2; twisttest(-17*2^(2*d), 2^(3*d), T2)

ellminimaltwist(ellinit([216,-432]),1)

a=ffgen(2^5,'a);ellinit([a]).j
a=ffgen(3^5,'a);ellinit([a]).j

ellfromeqn(y^3+(3+2*x)*y^2 +(6+5*x+4*x^2)*y+(10+9*x+8*x^2+7*x^3))
ellfromeqn(y^2+(2*x^2+3*x+4)*y+(5*x^4+6*x^3+7*x^2+8*x+9))
ellfromeqn(x^2+(2*y^2+3*y+4)*x+(5*y^4+6*y^3+7*y^2+8*y+9))
ellfromeqn((x^2+2*x+3)*y^2+(4*x^2+5*x+6)*y+(7*x^2+8*x+9))

e = ellinit([1,2,3,5,7]);
elldivpol(e,4)
elldivpol(e,-1)
elldivpol(e,-2)
elldivpol(e,-3)
ellxn(e,0)
ellxn(e,1)
ellxn(e,2)
ellxn(e,4)
ellxn(e,5)
P=[-1,1]; Q=ellmul(e,P,20);
if (ellisdivisible(e,Q,0,&A), A)
if (ellisdivisible(e,[0],0,&A), A)
if (ellisdivisible(e,Q,1,&A), A)
if (ellisdivisible(e,Q,20,&A), A)
if (ellisdivisible(e,Q,-20,&A), A)
if (ellisdivisible(e,Q,2,&A), A)
if (ellisdivisible(e,Q,5,&A), A)
if (ellisdivisible(e,Q, ellxn(e,5), &A), A)
if (ellisdivisible(e,Q,11,&A), A)
if (ellisdivisible(e,Q,22,&A), A)
e = ellinit([0,1]);
elldivpol(e,8)

\\ #2106
e = ellinit([0,0,1,-1,0]); P = [0,0];
K = nfinit(nfsplitting(elldivpol(e, 3)));
eK = ellinit([0,0,1,-1,0], K);
ellisdivisible(eK, P, 3)
P6 = ellmul(eK, P, 6);
if (ellisdivisible(eK, P6, 3, &Q), Q)
if (ellisdivisible(eK, P6, 2, &Q), Q)

e = ellinit([1.,2,3,4,5]);
e.eta
E = ellchangecurve(e, [2,3,4,5]);
E.eta
e = ellinit([1,2,3,4,5]); e.eta; e.roots;
E = ellchangecurve(e, [2,3,4,5]);
E.omega
E.eta
E.roots
ellglobalred(E)
ellglobalred(e)
p = 2^64+13;
E = ellchangecurve(e, [1/p, 2,3,4]);
elllocalred(E,p)

E = ellchangecurve(e, [1,0,0,0]);

E = ellchangecurve(e, [2,3,4,5]*Mod(1,7));
iferr(E.omega, e, e);
E.group

ellminimalmodel(ellinit([1/5,1/4,1/3,1/2,1],1),&v)
v

j=ffgen(2^5,'a);e = ellinit([j]); e.group; elllog(e,e.gen[1],e.gen[1])
E = ellchangecurve(e,[3,7,1,0]);
E.group

e = ellinit([Mod(1,3)]); e.group; elllog(e,e.gen[1],e.gen[1])
E = ellchangecurve(e,[2,7,1,0]);
E.group

e = ellinit([Mod(1,5)]); e.group; elllog(e,e.gen[1],e.gen[1])
E = ellchangecurve(e,[2,7,1,0]);
E.group

e = ellinit([1/3 +  O(3^9)]); e.roots
e = ellinit([1/3], O(3^5)); e.tate
iferr(ellinit(e,O(5)),E,E)
iferr(ellinit(e,5),E,E)
iferr(ellinit(e,1.0),E,E)
e.roots
E = ellchangecurve(e,[3,1,1,2]);
E.tate
ellztopoint(e,3)

e = ellinit([11/8], O(2^5));
e.tate

e = ellinit([1,0,1,4,-6], O(2^5));
e.tate

e = ellinit([0,-1,1,-10,-20]);
\\#1185
ellwp(e,x+O(x^12))
\\#1683
ellztopoint(e,3*e.omega[1]/5)
ellwp(e, 0.1 + 0.2*I, 1)

\\#1683
e = ellinit([1,1,0,-1740,22184]);
z=-3.0059315873096303229151114945365166621 + 0.E-36*I/2;
ellztopoint(e,z)

\\#1800
ellztopoint(ellinit([-1,0]), I)

\\#1186
ellmul(ellinit([3,0]), [1,2], -quadgen(-4))

\\#2060
ellzeta([1,I],2/10+I)
\\#2061
ellsigma([1,I],1/2+I)

logsigma(e,z='x)=if(type(z) != "t_POL" && type(z) != "t_SER", ellsigma(e,z,1));
v = [ellwp,ellzeta,ellsigma,logsigma];
ellperiods([1,0.1 + I/10^50])
e = ellinit([1,1]);
w = ellperiods([1,I/2])
w2 = ellperiods([1,I/2], 1)
u = [e,w,w2];
{
for (i = 1, #v,
  my(f = v[i]);
  for (j = 1, #u,
    my (a = u[j]);
    print([i,j]);
    print(f(a));
    print(f(a, x+O(x^10)));
    print(f(a, 1/3));
    print(f(a, I/3));
    print(f(a, (1+I)/3));
  )
)
}
ellK(1/3)
ellK(x+y); \\ would overflow stack
ellE(1/3)
ellK(1/3+I)
ellE(1/3+I)
elleta(e)
elleta([1,I])
v = [x->elleisnum(x,2),x->real(elleisnum(x,4,1)),x->real(elleisnum(x,6,1)),x->real(elleisnum(x,10))];
{
for (i = 1, #v,
  my(f = v[i]);
  print(f);
  print(f(e));
  print(f(w));
  print(f(w2));
)
}
elleisnum([1,I/2^30],2)
elleisnum([1,I/2^30],4)

\\ #2167
E=ellinit([0,-1,1,-10,-20]);
e=elleisnum(E,4,1)
bitprecision(e)

\\ #1257
ellrootno(ellinit([0,-1,1,217,-282]))

\\ #1296
e=ellinit([0,-1,0,-33,62]);
ellztopoint(e,-2.5261979245524788020279452840822073870+0.E-36*I)

\\ #1308
ellinit([108/91,11664/8281,-6561/8281,708588/753571,-14348907/68574961]).disc

e=ellinit([-4179962088,-104017551778988]);
P=[-12142738583336098442192836816176736488756409885083993663515903318775477283438352137135249323/325305271731471715644495647421121780765173058515779520952978859131039949432429351736409, 8334307521730470040507793911525024226184207596596408790562824419588749760136126836698728499174598419862301314492216896610149595/5867277807417841262595745323516111503774365731904549455908426496476272821377832232255239022995172589310668134318512865886323566173];
exponent(ellztopoint(e,ellpointtoz(e,P))[1] - P[1])



e=ellinit([1,3.+I]); x=1/2+I; y=ellordinate(e,x)[1];
ellztopoint(e,ellpointtoz(e,[x,y]))
testzellQp(e, P)=
{ my(a,q,Q);
  a = ellpointtoz(e,P);
  Q = ellztopoint(e,a);
  if (liftall(Q-P),error([e,P]));
  print(a);
  q = e.tate[3];
  a = a^2;
  if (type(a) == "t_PADIC",
    a /= q^(valuation(a,a.p) \ valuation(q,a.p)); /* reduce mod q^Z */
  );
  a - ellpointtoz(e, elladd(e,P,P));
}
e=ellinit([0,-1,1,0,0], O(11^5));
\\ BIB #1840
iferr(ellpointtoz(e,[Mod(3,'x^2+1),1]), E, E)
iferr(testzellQp(e, [0,0]), E, E)

e=ellinit([1,1,1,-10,-10], O(3^5));
testzellQp(e, [3,-2])
iferr(testzellQp(e, ellmul(e,[0,0],2)), E, E) \\ not on curve !

e=ellinit([2/9], O(3^10)); x=2; y=ellordinate(e,x)[1];
testzellQp(e, [x,y])

e=ellinit([1/4], O(2^10)); x=1/2; y=ellordinate(e,x)[1];
testzellQp(e, [x,y])

e=ellinit([1,-1,1,98,126],O(5^10));
testzellQp(e, [1,14])

e=ellinit([1,1,1,3,-5],O(2^10));
testzellQp(e,[5,10])

e=ellinit([1,-1,1,-48,147], O(2^10));
testzellQp(e,[13,-47])

ellinit([1,1], Mod(1,11))
ellrootno(ellinit([31^4,31^6]), 31)
e=ellinit([1,0,0,1,1]);
ellordinate(e, I)
E=ellchangecurve(e,[1/(2^4*3^4*5^2*7),2,3,4]);
forprime(p=2,11, if (ellap(e,p) != ellap(E,p),error(p)));
for(k=2,50, if (ellak(e,k) != ellak(E,k),error(k)));
if (ellan(e,100) != ellan(E,100),error("ellan"));
P=ellchangepoint([0,1],[1,2,3,4])
ellchangepointinv(P,[1,2,3,4])

\\#1416
E=ellinit([155818018413/16,-78179511999813417/32]);
ellminimalmodel(E,&v);
E2=ellchangecurve(E,v); ellminimalmodel(E2,&w); w

\\#1432
E=ellinit([-3,-60,480,0,0]); ellheight(E,[0,0])
iferr(ellheight(E,[0.,0]), e, e)

p=10^20+39; q=10^21+117;
e = ellinit([p^3*q^4,p^6*q^4]);
ellminimalmodel(e,&v); v
e = ellinit([p^4*q^4,p^6*q^4]);
ellminimalmodel(e,&v); v

e=ellminimalmodel(ellinit([1,1]));
e=ellchangecurve(e,1)
e=ellchangecurve(e,[2,0,0,0])
ellinit(ellinit([0,1]*Mod(1,5),ffgen(5^2)));

J=[0,1728,-3375,8000,54000,-32768,287496,-884736,-12288000,16581375,-884736000,-147197952000,-262537412640768000];
{
for (i=1,#J,
  my(e = ellinit([J[i]]));
  my(v = ellan(e,200));
  print("\n", e.j);
  forprime(p = 127, 200, print1(v[p]," "));
  print();
  forprime(p = 2^65, 2^65+1000, print1(ellap(e,p)," "));
);
}

p=2^32+15;
ellcard(ellinit([1,2,3,4,5], p))
E=ellinit([625,15625]);
elllocalred(E,5)
ellisoncurve(E,[0.,125.])
ellisoncurve(E,[1e-50,125+1e-50])
elladd(E,[0.,125.],[0.,125+1e-38])
iferr(ellmul([0,1,0,2,-15],[2,1],5),E,E)
x='x;
E=ellinit([x^2,x])
iferr(ellminimalmodel(E), e, e)
iferr(ellweilpairing(E,[0],[0],1), e, e)
ellinit([1,2],1.)
ellinit([ffgen(5),1],5)
ellinit([1.,1],precision(1.,60))

\\#1527
E = ellinit([0,0,0,-82,0]);
ellrootno(E,2)
ellrootno(E)
ellrootno(E,2)

E=ellinit([0,20,0,-352,512]);
ellrootno(E,2)
E.disc
elltamagawa(E)
ellbsd(E)

F=ellinit([-1456/243,93184/19683]); /* isomorphic to E */
ellrootno(F,2)
F.disc
elltamagawa(F)
ellbsd(F)

E=ellinit([1,0,1,4,-6]);
E.disc
elltamagawa(E)
ellbsd(E)

ellbsd(ellinit([-5*36^4,0]))

\\#1558
ellap(ellinit([-1137195,489565862]),2038074751)
ellap(ellinit([582304190,64196421]),2147438927)
e=ellinit([0,-1,1,-2,2]); P=[2,-2]; ellnonsingularmultiple(e, P)
e=ellinit([-111,214]); P=[35,-198]; v = [4*3,12,214,12];
e=ellchangecurve(e,v); P=ellchangepoint(P,v);
ellnonsingularmultiple(e,P)
e=ellinit([0,-1,1,-2,2]);
test(f)=print(f(e,,'t)); print(f(e,5)); print(f(e,5,'t));
F=[ellformalw,ellformalpoint,ellformaldifferential,ellformallog,ellformalexp];
for(i=1,#F,print(F[i]);test(F[i]));

\\#1637
U=[-245369975623514803521420211420402474815373720177518420696215292670924086780233779586506792145074717/4069861969699146045958032034788970995470603578576,-696053034387689435413673877713244121475782562610666903528251447608130654196417305610237634015759861/4069861969699146045958032034788970995470603578576,-49356694805693369953216822105035107974083562942605713355717183776373331122267882948881129025418396548321363993080002121985116427859996985649385099335/4069861969699146045958032034788970995470603578576,5996724761485351384753083917633373954904218466995628499407992841502416127074022935350022919212488198205525833829916828282709837728834438536000230267/1017465492424786511489508008697242748867650894644,-1943033527672518505867347410801184866534773028979571976479290969993491996577167508758887358085512124817167326269402804641328588524702957475568608101932310746684310251168033629279636817525659690186705/4069861969699146045958032034788970995470603578576];
E=ellinit(U);
P=[283188771261789732541662405743402980051199109235,39175522019721182331498476245534716069785193060995];
ellheight(E,P)

\\#1648
ellinit([0,0,0,0,1]).a1

E=ellinit([-22032-15552*x,0], nfinit(x^2-2)); P=[-72*x-108,0];
ellpointtoz(E,P)

E=ellinit([134199387,91266697],3031140653);
P = [911071793,1086747416]; Q = ellmul(E,P,2);
elllog(E,Q,P)

ellheight(ellinit([1,0,1,4,-6]))

E=ellinit([0,1,1,20,-890]);
P=ellmul(E,[16,61],45*23);
ellsaturation(E,[P],100)

E=ellinit([0,0,1,-7,6]);
V=[[1879462415967166/931705846156449,-30751668616480047298756/28439256014143448903793],[-4918796969449106250916/1682155121549140256041,-124886419810479084631920999334858/68992052177891195733628251285589],[72257627402076649/69972081056148496,-2774239841245604882535253/18509180294972016896964544]];
ellsaturation(E,V,100)
E=ellinit([1,0,0,-302,-2036]);
P=ellmul(E,[-10,8],27);
ellsaturation(E,[P],100)
E=ellinit([1,0,0,-24432,-1471934]);
P=ellmul(E,[-36131/400,361313/8000],27);
ellsaturation(E,[P],100)

\\ ERRORS, keep at end of file
ellK(1)
ellinit([])
ellinit([1,1],quadgen(5))
ellinit([Mod(1,2),1],O(2))
ellinit([O(2),1],ffgen(2^3))
ellinit([O(2),1],1.)
ellinit([ffgen(5),1],3)
ellinit([1.,Mod(1,3)])
e=ellinit([0,-1,1,-2,2]);
ellformalw(e,0)
ellformalw(e,-1)
elltwist([0,0])