Q = [2,1;1,2];
qfeval(Q,[x,y])
 [x,y]*Q*[x,y]~
qfeval(Q,[x1,y1],[x2,y2])
 [x1,y1]*Q*[x2,y2]~

R = [1,2,3;2,4,2;3,2,5];
matdet(R)
qfsign(R)
[U,V]=qfgaussred(R,1)
 U~*matdiagonal(V)*U == R

[L, V]=qfjacobi(Q)
norml2(V~*Q*V - matdiagonal(L))
V~*V==1

L = qfcholesky(Q)
L~*L==Q

L = [1,0,0;0,1,0;4,5,6];
[q,r]=matqr(L);
q
r
norml2(q*r-L)

M = R~*R
qfsign(M)
T = qflllgram(M)
T~*M*T

L = qfcholesky(M);
S = qflll(L)
S == T
round((L*S)~*(L*S))

z7 = zeta(7); z25=zeta(2)*zeta(5);
z43 = zeta(4)*zeta(3);
lindep([zetamult([3,2,2]),z7,z25,z43])
(157*z7-120*z25+36*z43)/16 - zetamult([3,2,2])

qfminim(Q)

qfminim(Q,10)

qfcvp(Q,[10/3,10/3])

qfrep(Q,20)
L = lfunqf(Q); lfunan(L,20)
lfunparams(L)
lfun(L,1) - 2*Pi/sqrt(3)/x
[mf,F,v] = mffromqf(Q); mfcoefs(F,20)
mfparams(F)

p = x^6+6*y*x^5-20*y^3*x^3-15*y^4*x^2+y^6;
p'' - deriv(p',y) + derivn(p,2,y)
[mf2,F2,v2] = mffromqf(Q, p); mfcoefs(F2,20)
mfparams(F2)

qfperfection(Q)

qfsolve([1,0,0;0,3,0;0,0,-21])

M = qfparam([1,0,0;0,3,0;0,0,-21],[3,2,1]~)
v = y^2 * M*[1,x/y,(x/y)^2]~
v[1]^2+3*v[2]^2-21*v[3]^2

G = matdiagonal([650, -104329, -104329]);
[H,U]=qfminimize(G); H
U
U~*G*U

qfauto(matid(3))
K=nfinit(x^3-3*x+1); L=round(K.t2)
qfauto(L)
T=qflllgram(L); M = T~*L*T; qfisom(L,M)
Q=qfisominit(L); qfisom(Q,M)

 Q=matid(6); G=qfauto(Q); V=qfminim(Q,3);
 apply(x->[x[1],#x],qforbits(G,V))

E8 = matrix(8,8,i,j,if(i==1&&j==1,4, \
       i==j || (i==1 && j<8) || (j==1 && i<8),2,1));
E8==E8~ \\ symmetric
matdet(E8) \\ unimodular
qfsign(E8) \\ signature
L = qfminim(E8); L[1..2] \\ 240 minimal vectors
V = L[3][,1] \\ first minimal vector
qfeval(E8,V) \\ the norm is 2

qfperfection(E8) \\ perfection rank
G=qfauto(E8); G[1] \\ number of isometries
A=G[2][1] \\ one isomorphim
A~*E8*A==E8

[mf,F,C]=mffromqf(E8);
mfparams(F)
mfcoefs(F,10)
mfcoef(F,100003)
L = lfunqf(E8);
lfunparams(L)
lfun(L,0)

V=concat([vector(23,i,2*i+1),51,145]);
K=matkerint(Mat(V));
M=matdiagonal(vector(25,i,if(i==25,-1,1)));
L24 = K~*M*K; \\ Leech lattice
matdet(L24) \\ unimodular
L = qfminim(L24); L[1..2]
qfperfection(L24)

G=qfauto(L24); G[1] \\ number of isometries
[mf,F,C]=mffromqf(L24);
mfparams(F)
mfcoef(F,100003)