Clean up PEGASIS submodule inclusion

4 files changed, 0 insertions, 1353 deletions

diff --git a/theta_lib/basis_change/base_change_dim2.py b/theta_lib/basis_change/base_change_dim2.py
deleted file mode 100644
index 4994529..0000000
--- a/theta_lib/basis_change/base_change_dim2.py
+++ /dev/null
@@ -1,150 +0,0 @@
-from sage.all import *
-
-import itertools
-
-def complete_symplectic_matrix_dim2(C, D, n=4):
-	Zn = Integers(n)
-
-	# Compute first row
-	F = D.stack(-C).transpose()
-	xi = F.solve_right(vector(Zn, [1, 0]))
-
-	# Rearrange TODO: why?
-	v = vector(Zn, [xi[2], xi[3], -xi[0], -xi[1]])
-
-	# Compute second row
-	F2 = F.stack(v)
-	yi = F2.solve_right(vector(Zn, [0, 1, 0]))
-	M = Matrix(Zn,
-		[
-			[xi[0], yi[0], C[0, 0], C[0, 1]],
-			[xi[1], yi[1], C[1, 0], C[1, 1]],
-			[xi[2], yi[2], D[0, 0], D[0, 1]],
-			[xi[3], yi[3], D[1, 0], D[1, 1]],
-		],
-	)
-
-	return M
-
-def block_decomposition(M):
-	"""
-	Given a 4x4 matrix, return the four 2x2 matrices as blocks
-	"""
-	A = M.matrix_from_rows_and_columns([0, 1], [0, 1])
-	B = M.matrix_from_rows_and_columns([2, 3], [0, 1])
-	C = M.matrix_from_rows_and_columns([0, 1], [2, 3])
-	D = M.matrix_from_rows_and_columns([2, 3], [2, 3])
-	return A, B, C, D
-
-def is_symplectic_matrix_dim2(M):
-	A, B, C, D = block_decomposition(M)
-	if B.transpose() * A != A.transpose() * B:
-		return False
-	if C.transpose() * D != D.transpose() * C:
-		return False
-	if A.transpose() * D - B.transpose() * C != identity_matrix(2):
-		return False
-	return True
-
-def base_change_theta_dim2(M, zeta):
-    r"""
-    Computes the matrix N (in row convention) of the new level 2
-    theta coordinates after a symplectic base change of A[4] given
-    by M\in Sp_4(Z/4Z). We have:
-    (theta'_i)=N*(theta_i)
-    where the theta'_i are the new theta coordinates and theta_i,
-    the theta coordinates induced by the product theta structure.
-    N depends on the fourth root of unity zeta=e_4(Si,Ti) (where
-    (S0,S1,T0,T1) is a symplectic basis if A[4]).
-
-    Inputs:
-    - M: symplectic base change matrix.
-    - zeta: a primitive 4-th root of unity induced by the Weil-pairings
-    of the symplectic basis of A[4].
-
-    Output: Matrix N of base change of theta-coordinates.
-    """
-    # Split 4x4 matrix into 2x2 blocks
-    A, B, C, D = block_decomposition(M)
-
-    # Initialise N to 4x4 zero matrix
-    N = [[0 for _ in range(4)] for _ in range(4)]
-
-    def choose_non_vanishing_index(C, D, zeta):
-        """
-        Choice of reference non-vanishing index (ir0,ir1)
-        """
-        for ir0, ir1 in itertools.product([0, 1], repeat=2):
-            L = [0, 0, 0, 0]
-            for j0, j1 in itertools.product([0, 1], repeat=2):
-                k0 = C[0, 0] * j0 + C[0, 1] * j1
-                k1 = C[1, 0] * j0 + C[1, 1] * j1
-
-                l0 = D[0, 0] * j0 + D[0, 1] * j1
-                l1 = D[1, 0] * j0 + D[1, 1] * j1
-
-                e = -(k0 + 2 * ir0) * l0 - (k1 + 2 * ir1) * l1
-                L[ZZ(k0 + ir0) % 2 + 2 * (ZZ(k1 + ir1) % 2)] += zeta ** (ZZ(e))
-
-            # Search if any L value in L is not zero
-            if any([x != 0 for x in L]):
-                return ir0, ir1
-
-    ir0, ir1 = choose_non_vanishing_index(C, D, zeta)
-
-    for i0, i1, j0, j1 in itertools.product([0, 1], repeat=4):
-        k0 = A[0, 0] * i0 + A[0, 1] * i1 + C[0, 0] * j0 + C[0, 1] * j1
-        k1 = A[1, 0] * i0 + A[1, 1] * i1 + C[1, 0] * j0 + C[1, 1] * j1
-
-        l0 = B[0, 0] * i0 + B[0, 1] * i1 + D[0, 0] * j0 + D[0, 1] * j1
-        l1 = B[1, 0] * i0 + B[1, 1] * i1 + D[1, 0] * j0 + D[1, 1] * j1
-
-        e = i0 * j0 + i1 * j1 - (k0 + 2 * ir0) * l0 - (k1 + 2 * ir1) * l1
-        N[i0 + 2 * i1][ZZ(k0 + ir0) % 2 + 2 * (ZZ(k1 + ir1) % 2)] += zeta ** (ZZ(e))
-
-    return Matrix(N)
-
-def montgomery_to_theta_matrix_dim2(zero12,N=identity_matrix(4),return_null_point=False):
-	r"""
-	Computes the matrix that transforms Montgomery coordinates on E1*E2 into theta coordinates 
-	with respect to a certain theta structure given by a base change matrix N.
-
-	Input: 
-	- zero12: theta-null point for the product theta structure on E1*E2[2]:
-	zero12=[zero1[0]*zero2[0],zero1[1]*zero2[0],zero1[0]*zero2[1],zero1[1]*zero2[1]],
-	where the zeroi are the theta-null points of Ei for i=1,2.
-	- N: base change matrix from the product theta-structure.
-	- return_null_point: True if the theta-null point obtained after applying N to zero12 is returned.
-
-	Output:
-	- Matrix for the change of coordinates:
-	(X1*X2,Z1*X2,X1*Z2,Z1*Z2)-->(theta_00,theta_10,theta_01,theta_11).
-	- If return_null_point, the theta-null point obtained after applying N to zero12 is returned.
-	"""
-
-	Fp2=zero12[0].parent()
-
-	M=zero_matrix(Fp2,4,4)
-
-	for i in range(4):
-		for j in range(4):
-			M[i,j]=N[i,j]*zero12[j]
-
-	M2=[]
-	for i in range(4):
-		M2.append([M[i,0]+M[i,1]+M[i,2]+M[i,3],-M[i,0]-M[i,1]+M[i,2]+M[i,3],-M[i,0]+M[i,1]-M[i,2]+M[i,3],M[i,0]-M[i,1]-M[i,2]+M[i,3]])
-
-	if return_null_point:
-		null_point=[M[i,0]+M[i,1]+M[i,2]+M[i,3] for i in range(4)]
-		return Matrix(M2), null_point
-	else:
-		return Matrix(M2)
-
-def apply_base_change_theta_dim2(N,P):
-	Q=[]
-	for i in range(4):
-		Q.append(0)
-		for j in range(4):
-			Q[i]+=N[i,j]*P[j]
-	return Q
-
diff --git a/theta_lib/basis_change/base_change_dim4.py b/theta_lib/basis_change/base_change_dim4.py
deleted file mode 100644
index aaf685f..0000000
--- a/theta_lib/basis_change/base_change_dim4.py
+++ /dev/null
@@ -1,152 +0,0 @@
-from sage.all import *
-import itertools
-
-from ..theta_structures.theta_helpers_dim4 import multindex_to_index
-
-def bloc_decomposition(M):
-	I1=[0,1,2,3]
-	I2=[4,5,6,7]
-
-	A=M[I1,I1]
-	B=M[I2,I1]
-	C=M[I1,I2]
-	D=M[I2,I2]
-
-	return A,B,C,D
-
-def mat_prod_vect(A,I):
-	J=[]
-	for i in range(4):
-		J.append(0)
-		for k in range(4):
-			J[i]+=A[i,k]*I[k]
-	return tuple(J)
-
-def add_tuple(I,J):
-	K=[]
-	for k in range(4):
-		K.append(I[k]+J[k])
-	return tuple(K)
-
-def scal_prod_tuple(I,J):
-	s=0
-	for k in range(4):
-		s+=I[k]*J[k]
-	return s
-
-def red_mod_2(I):
-	J=[]
-	for x in I:
-		J.append(ZZ(x)%2)
-	return tuple(J)
-
-def choose_non_vanishing_index(C,D,zeta):
-	for I0 in itertools.product([0,1],repeat=4):
-		L=[0 for k in range(16)]
-		for J in itertools.product([0,1],repeat=4):
-			CJ=mat_prod_vect(C,J)
-			DJ=mat_prod_vect(D,J)
-			e=-scal_prod_tuple(CJ,DJ)-2*scal_prod_tuple(I0,DJ)
-
-			I0pDJ=add_tuple(I0,CJ)
-
-			L[multindex_to_index(red_mod_2(I0pDJ))]+=zeta**(ZZ(e))
-		for k in range(16):
-			if L[k]!=0:
-				return I0,L
-
-
-def base_change_theta_dim4(M,zeta):
-	
-	Z4=Integers(4)
-	A,B,C,D=bloc_decomposition(M.change_ring(Z4))
-
-	I0,L0=choose_non_vanishing_index(C,D,zeta)
-
-
-	N=[L0]+[[0 for j in range(16)] for i in range(15)]
-	for I in itertools.product([0,1],repeat=4):
-		if I!=(0,0,0,0):
-			AI=mat_prod_vect(A,I)
-			BI=mat_prod_vect(B,I)
-			for J in itertools.product([0,1],repeat=4):
-				CJ=mat_prod_vect(C,J)
-				DJ=mat_prod_vect(D,J)
-
-				AIpCJ=add_tuple(AI,CJ)
-				BIpDJ=add_tuple(BI,DJ)
-
-				e=scal_prod_tuple(I,J)-scal_prod_tuple(AIpCJ,BIpDJ)-2*scal_prod_tuple(I0,BIpDJ)
-				N[multindex_to_index(I)][multindex_to_index(red_mod_2(add_tuple(AIpCJ,I0)))]+=zeta**(ZZ(e))
-
-	Fp2=zeta.parent()
-	return matrix(Fp2,N)
-
-def apply_base_change_theta_dim4(N,P):
-	Q=[]
-	for i in range(16):
-		Q.append(0)
-		for j in range(16):
-			Q[i]+=N[i,j]*P[j]
-	return Q
-
-def complete_symplectic_matrix_dim4(C,D,n=4):
-	Zn=Integers(n)
-
-	Col_I4=[matrix(Zn,[[1],[0],[0],[0]]),matrix(Zn,[[0],[1],[0],[0],[0]]),
-	matrix(Zn,[[0],[0],[1],[0],[0],[0]]),matrix(Zn,[[0],[0],[0],[1],[0],[0],[0]])]
-
-	L_DC=block_matrix([[D.transpose(),-C.transpose()]])
-	Col_AB_i=L_DC.solve_right(Col_I4[0])
-	A_t=Col_AB_i[[0,1,2,3],0].transpose()
-	B_t=Col_AB_i[[4,5,6,7],0].transpose()
-
-	for i in range(1,4):
-		F=block_matrix(2,1,[L_DC,block_matrix(1,2,[B_t,-A_t])])
-		Col_AB_i=F.solve_right(Col_I4[i])
-		A_t=block_matrix(2,1,[A_t,Col_AB_i[[0,1,2,3],0].transpose()])
-		B_t=block_matrix(2,1,[B_t,Col_AB_i[[4,5,6,7],0].transpose()])
-
-	A=A_t.transpose()
-	B=B_t.transpose()
-
-	M=block_matrix([[A,C],[B,D]])
-
-	return M
-
-def random_symmetric_matrix(n,r):
-	Zn=Integers(n)
-	M=zero_matrix(Zn,r,r)
-	for i in range(r):
-		M[i,i]=randint(0,n-1)
-	for i in range(r):
-		for j in range(i+1,r):
-			M[i,j]=randint(0,n-1)
-			M[j,i]=M[i,j]
-	return M
-
-
-def random_symplectic_matrix(n):
-	Zn=Integers(n)
-
-	C=random_symmetric_matrix(n,4)
-	Cp=random_symmetric_matrix(n,4)
-
-	A=random_matrix(Zn,4,4)
-	while not A.is_invertible():
-		A=random_matrix(Zn,4,4)
-
-	tA_inv=A.inverse().transpose()
-	ACp=A*Cp
-	return block_matrix([[ACp,A],[C*ACp-tA_inv,C*A]])
-
-def is_symplectic_matrix_dim4(M):
-	A,B,C,D=bloc_decomposition(M)
-	if B.transpose()*A!=A.transpose()*B:
-		return False
-	if C.transpose()*D!=D.transpose()*C:
-		return False
-	if A.transpose()*D-B.transpose()*C!=identity_matrix(4):
-		return False
-	return True
-
diff --git a/theta_lib/basis_change/canonical_basis_dim1.py b/theta_lib/basis_change/canonical_basis_dim1.py
deleted file mode 100644
index e1c3d1f..0000000
--- a/theta_lib/basis_change/canonical_basis_dim1.py
+++ /dev/null
@@ -1,76 +0,0 @@
-from sage.all import *
-from ..utilities.discrete_log import weil_pairing_pari, discrete_log_pari
-
-def last_four_torsion(E):
-	a_inv=E.a_invariants()
-	A =a_inv[1]
-	if a_inv != (0,A,0,1,0):
-		raise ValueError("The elliptic curve E is not in the Montgomery model.")
-	y2=A-2
-	y=y2.sqrt()
-	return E([-1,y,1])
-
-
-def make_canonical(P,Q,A,preserve_pairing=False):
-	r"""
-	Input:
-	- P,Q: a basis of E[A].
-	- A: an integer divisible by 4.
-	- preserve_pairing: boolean indicating if we want to preserve pairing at level 4.
-	
-	Output:
-	- P1,Q1: basis of E[A].
-	- U1,U2: basis of E[4] induced by (P1,Q1) ((A//4)*P1=U1, (A//4)*Q1=U2) such that U2[0]=-1 
-	and e_4(U1,U2)=i if not preserve_pairing and e_4(U1,U2)=e_4((A//4)*P,(A//4)*Q) if preserve_pairing.
-	- M: base change matrix (in row convention) from (P1,Q1) to (P,Q).
-	
-	We say that (U1,U2) is canonical and that (P1,Q1) induces or lies above a canonical basis. 
-	"""
-	E=P.curve()
-	Fp2=E.base_ring()
-	i=Fp2.gen()
-
-	assert i**2==-1
-
-	T2=last_four_torsion(E)
-	V1=(A//4)*P
-	V2=(A//4)*Q
-	U1=V1
-	U2=V2
-	
-	a1=discrete_log_pari(weil_pairing_pari(U1,T2,4),i,4)
-	b1=discrete_log_pari(weil_pairing_pari(U2,T2,4),i,4)
-
-	if a1%2!=0:
-		c1=inverse_mod(a1,4)
-		d1=c1*b1
-		P1=P
-		Q1=Q-d1*P
-		U1,U2=U1,U2-d1*U1
-		M=matrix(ZZ,[[1,0],[d1,1]])
-	else:
-		c1=inverse_mod(b1,4)
-		d1=c1*a1
-		P1=Q
-		Q1=P-d1*Q
-		U1,U2=U2,U1-d1*U2
-		M=matrix(ZZ,[[d1,1],[1,0]])
-
-	if preserve_pairing:
-		e4=weil_pairing_pari(V1,V2,4)
-	else:
-		e4=i
-	
-	if weil_pairing_pari(U1,U2,4)!=e4:
-		U2=-U2
-		Q1=-Q1
-		M[0,1]=-M[0,1]
-		M[1,1]=-M[1,1]
-	
-	assert (A//4)*P1==U1
-	assert (A//4)*Q1==U2
-	assert weil_pairing_pari(U1,U2,4)==e4
-	assert M[0,0]*P1+M[0,1]*Q1==P
-	assert M[1,0]*P1+M[1,1]*Q1==Q
-	
-	return P1,Q1,U1,U2,M
diff --git a/theta_lib/basis_change/kani_base_change.py b/theta_lib/basis_change/kani_base_change.py
deleted file mode 100644
index e6de2e2..0000000
--- a/theta_lib/basis_change/kani_base_change.py
+++ /dev/null
@@ -1,975 +0,0 @@
-from sage.all import *
-from ..basis_change.canonical_basis_dim1 import make_canonical
-from ..basis_change.base_change_dim2 import is_symplectic_matrix_dim2
-from ..basis_change.base_change_dim4 import (
-    complete_symplectic_matrix_dim4,
-    is_symplectic_matrix_dim4,
-    bloc_decomposition,
-)
-from ..theta_structures.Tuple_point import TuplePoint
-
-
-def base_change_canonical_dim2(P1,P2,R1,R2,q,f):
-	r"""
-
-	Input:
-	- P1, P2: basis of E1[2**f].
-	- R1, R2: images of P1, P2 by \sigma: E1 --> E2.
-	- q: degree of \sigma.
-	- f: log_2(order of P1 and P2).
-
-	Output:
-	- P1_doubles: list of 2**i*P1 for i in {0,...,f-2}.
-	- P2_doubles: list of 2**i*P2 for i in {0,...,f-2}.
-	- R1_doubles: list of 2**i*R1 for i in {0,...,f-2}.
-	- R2_doubles: list of 2**i*R2 for i in {0,...,f-2},
-	- T1, T2: canonical basis of E1[4].
-	- U1, U2: canonical basis of E2[4].
-	- M0: base change matrix of the symplectic basis 2**(f-2)*B1 of E1*E2[4] given by:
-	B1:=[[(P1,0),(0,R1)],[(P2,0),(0,lamb*R2)]]
-	where lamb is the modular inverse of q mod 2**f, so that:
-	e_{2**f}(P1,P2)=e_{2**f}(R1,lamb*R2).
-	in the canonical symplectic basis:
-	B0:=[[(T1,0),(0,U1)],[(T2,0),(0,U2)]].
-	"""
-	lamb=inverse_mod(q,4)
-
-	P1_doubles=[P1]
-	P2_doubles=[P2]
-	R1_doubles=[R1]
-	R2_doubles=[R2]
-
-	for i in range(f-2):
-		P1_doubles.append(2*P1_doubles[-1])
-		P2_doubles.append(2*P2_doubles[-1])
-		R1_doubles.append(2*R1_doubles[-1])
-		R2_doubles.append(2*R2_doubles[-1])
-
-	# Constructing canonical basis of E1[4] and E2[4].
-	_,_,T1,T2,MT=make_canonical(P1_doubles[-1],P2_doubles[-1],4,preserve_pairing=True)
-	_,_,U1,U2,MU=make_canonical(R1_doubles[-1],lamb*R2_doubles[-1],4,preserve_pairing=True)
-
-	Z4=Integers(4)
-	M0=matrix(Z4,[[MT[0,0],0,MT[1,0],0],
-		[0,MU[0,0],0,MU[1,0]],
-		[MT[0,1],0,MT[1,1],0],
-		[0,MU[0,1],0,MU[1,1]]])
-
-	return P1_doubles,P2_doubles,R1_doubles,R2_doubles,T1,T2,U1,U2,M0
-
-def gluing_base_change_matrix_dim2(a1,a2,q):
-	r"""Computes the symplectic base change matrix of a symplectic basis (*,B_K4) of E1*E2[4]
-	given by the kernel of the dimension 2 gluing isogeny:
-	B_K4=2**(f-2)[([a1]P1-[a2]P2,R1),([a1]P2+[a2]P1,R2)]
-	in the basis $2**(f-2)*B1$ given by:
-	B1:=[[(P1,0),(0,R1)],[(P2,0),(0,lamb*R2)]]
-	where:
-	- lamb is the inverse of q modulo 2**f.
-	- (P1,P2) is the canonical basis of E1[2**f].
-	- (R1,R2) is the image of (P1,P2) by sigma.
-
-	Input:
-	- a1, q: integers.
-
-	Output:
-	- M: symplectic base change matrix of (*,B_K4) in 2**(f-2)*B1.
-	"""
-
-	Z4=Integers(4)
-
-	mu=inverse_mod(a1,4)
-
-	A=matrix(Z4,[[0,mu],
-		[0,0]])
-	B=matrix(Z4,[[0,0],
-		[-1,-ZZ(mu*a2)]])
-
-	C=matrix(Z4,[[ZZ(a1),ZZ(a2)],
-		[1,0]])
-	D=matrix(Z4,[[-ZZ(a2),ZZ(a1)],
-		[0,ZZ(q)]])
-
-	#M=complete_symplectic_matrix_dim2(C, D, 4)
-	M=block_matrix([[A,C],[B,D]])
-
-	assert is_symplectic_matrix_dim2(M)
-
-	return M
-
-# ============================================== #
-#     Functions for the class KaniClapotiIsog    #
-# ============================================== #
-
-def clapoti_cob_matrix_dim2(integers):
-	gu,xu,yu,gv,xv,yv,Nbk,Nck,e,m = integers
-
-	xu = ZZ(xu)
-	xv = ZZ(xv)
-	Nbk = ZZ(Nbk)
-	Nck = ZZ(Nck)
-	u = ZZ(gu*(xu**2+yu**2))
-	v = ZZ(gv*(xv**2+yv**2))
-	mu = inverse_mod(u,4)
-	suv = xu*xv+yu*yv
-	inv_Nbk = inverse_mod(Nbk,4)
-	inv_gugvNcksuv = inverse_mod(gu*gv*Nck*suv,4)
-
-	Z4=Integers(4)
-
-	M=matrix(Z4,[[0,0,u*Nbk,0],
-		[0,inv_Nbk*inv_gugvNcksuv,gu*suv,0],
-		[-inv_Nbk*mu,0,0,gu*Nbk*u],
-		[0,0,0,gu*gv*Nbk*Nck*suv]])
-
-	assert is_symplectic_matrix_dim2(M)
-
-	return M
-
-def clapoti_cob_matrix_dim2_dim4(integers):
-	gu,xu,yu,gv,xv,yv,Nbk,Nck,e,m = integers
-
-	xu = ZZ(xu)
-	yu = ZZ(yu)
-	xv = ZZ(xv)
-	yv = ZZ(yv)
-	gu = ZZ(gu)
-	gv = ZZ(gv)
-	Nbk = ZZ(Nbk)
-	Nck = ZZ(Nck)
-	u = ZZ(gu*(xu**2+yu**2))
-	v = ZZ(gv*(xv**2+yv**2))
-	suv = xu*xv+yu*yv
-	duv = xv*yu-xu*yv
-	duv_2m = duv//2**m
-	mu = inverse_mod(u,4)
-	nu = inverse_mod(v,4)
-	sigmauv = inverse_mod(suv,4)
-	inv_guNbk = inverse_mod(gu*Nbk,4)
-	lamb = nu*gu*gv*Nbk*suv
-	mu1 = ZZ(mu*gu**2*gv*suv*Nbk*Nck*duv_2m)
-	mu2 = ZZ(duv_2m*gu*sigmauv*(Nbk*u*yu+gv*xv*Nck*duv))
-	mu3 = ZZ(duv_2m*gu*sigmauv*(Nbk*u*xu-gv*yv*Nck*duv))
-
-	Z4=Integers(4)
-
-	M=matrix(Z4,[[gu*xu,-gu*yu,0,0,0,0,mu2,mu3],
-		[0,0,lamb*xv,-lamb*yv,mu1*yu,mu1*xu,0,0],
-		[gu*yu,gu*xu,0,0,0,0,-mu3,mu2],
-		[0,0,lamb*yv,lamb*xv,-mu1*xu,mu1*yu,0,0],
-		[0,0,0,0,mu*xu,-mu*yu,0,0],
-		[0,0,0,0,0,0,inv_guNbk*xv*sigmauv,-inv_guNbk*yv*sigmauv],
-		[0,0,0,0,mu*yu,mu*xu,0,0],
-		[0,0,0,0,0,0,inv_guNbk*yv*sigmauv,inv_guNbk*xv*sigmauv]])
-
-	assert is_symplectic_matrix_dim4(M)
-
-	return M
-
-def clapoti_cob_splitting_matrix(integers):
-	gu,xu,yu,gv,xv,yv,Nbk,Nck,e,m = integers
-
-	v=ZZ(gv*(xv**2+yv**2))
-	vNck=ZZ(v*Nck)
-	inv_vNck=inverse_mod(vNck,4)
-
-	Z4=Integers(4)
-
-	M=matrix(Z4,[[0,0,0,0,-1,0,0,0],
-		[0,0,0,0,0,-1,0,0],
-		[0,0,vNck,0,0,0,0,0],
-		[0,0,0,vNck,0,0,0,0],
-		[1,0,-vNck,0,0,0,0,0],
-		[0,1,0,-vNck,0,0,0,0],
-		[0,0,0,0,1,0,inv_vNck,0],
-		[0,0,0,0,0,1,0,inv_vNck]])
-
-	assert is_symplectic_matrix_dim4(M)
-
-	return M
-
-# =============================================== #
-#     Functions for the class KaniFixedDegDim2    #
-# =============================================== #
-
-def fixed_deg_gluing_matrix_Phi1(u,a,b,c,d):
-	u,a,b,c,d = ZZ(u),ZZ(a),ZZ(b),ZZ(c),ZZ(d)
-
-	mu = inverse_mod(u,4)
-	inv_cmd = inverse_mod(c-d,4)
-
-	Z4 = Integers(4)
-
-	M = matrix(Z4,[[0,0,u,0],
-		[0,inv_cmd,c+d,0],
-		[-mu,0,0,(d**2-c**2)*mu],
-		[0,0,0,c-d]])
-
-	assert is_symplectic_matrix_dim2(M)
-
-	return M
-
-def fixed_deg_gluing_matrix_Phi2(u,a,b,c,d):
-	u,a,b,c,d = ZZ(u),ZZ(a),ZZ(b),ZZ(c),ZZ(d)
-
-	mu = inverse_mod(u,4)
-	inv_cpd = inverse_mod(c+d,4)
-
-	Z4 = Integers(4)
-
-	M = matrix(Z4,[[0,0,u,0],
-		[0,-inv_cpd,d-c,0],
-		[-mu,0,0,(d**2-c**2)*mu],
-		[0,0,0,-(c+d)]])
-
-	assert is_symplectic_matrix_dim2(M)
-
-	return M
-
-def fixed_deg_gluing_matrix_dim4(u,a,b,c,d,m):
-	u,a,b,c,d = ZZ(u),ZZ(a),ZZ(b),ZZ(c),ZZ(d)
-
-	mu = inverse_mod(u,4)
-	nu = ZZ((-mu**2)%4)
-	amb_2m = ZZ((a-b)//2**m)
-	apb_2m = ZZ((a+b)//2**m)
-	u2pc2md2_2m = ZZ((u**2+c**2-d**2)//2**m)
-	inv_cmd = inverse_mod(c-d,4)
-	inv_cpd = inverse_mod(c+d,4)
-
-
-	Z4 = Integers(4)
-
-	M = matrix(Z4,[[1,0,0,0,0,0,-u2pc2md2_2m,-apb_2m*(c+d)],
-		[0,0,(c+d)*(c-d)*nu,(a-b)*(c-d)*nu,0,amb_2m*(c-d),0,0],
-		[0,1,0,0,0,0,amb_2m*(c-d),-u2pc2md2_2m],
-		[0,0,-(a+b)*(c+d)*nu,(c+d)*(c-d)*nu,-apb_2m*(c+d),0,0,0],
-		[0,0,0,0,1,0,0,0],
-		[0,0,0,0,0,0,1,(a+b)*inv_cmd],
-		[0,0,0,0,0,1,0,0],
-		[0,0,0,0,0,0,(b-a)*inv_cpd,1]])
-
-	assert is_symplectic_matrix_dim4(M)
-
-	return M
-
-def fixed_deg_gluing_matrix(u,a,b,c,d):
-	r"""
-	Deprecated.
-	"""
-
-	mu = inverse_mod(u,4)
-	nu = (-mu**2)%4
-
-	Z4 = Integers(4)
-
-	M = matrix(Z4,[[0,0,0,0,ZZ(u),0,0,0],
-		[0,0,0,0,0,ZZ(u),0,0],
-		[0,0,ZZ(nu*(a+b)),ZZ(nu*(d-c)),ZZ(a+b),ZZ(d-c),0,0],
-		[0,0,ZZ(nu*(c+d)),ZZ(nu*(a-b)),ZZ(c+d),ZZ(a-b),0,0],
-		[ZZ(-mu),0,0,0,0,0,ZZ(u),0],
-		[0,ZZ(-mu),0,0,0,0,0,ZZ(u)],
-		[0,0,0,0,0,0,ZZ(a-b),ZZ(-c-d)],
-		[0,0,0,0,0,0,ZZ(c-d),ZZ(a+b)]])
-
-	assert is_symplectic_matrix_dim4(M)
-
-	return M
-
-def fixed_deg_splitting_matrix(u):
-
-	mu = inverse_mod(u,4)
-
-	Z4 = Integers(4)
-
-	M = matrix(Z4,[[0,0,0,0,-1,0,0,0],
-		[0,0,0,0,0,-1,0,0],
-		[0,0,ZZ(-u),0,0,0,0,0],
-		[0,0,0,ZZ(-u),0,0,0,0],
-		[1,0,ZZ(-mu),0,0,0,0,0],
-		[0,1,0,ZZ(-mu),0,0,0,0],
-		[0,0,0,0,ZZ(mu),0,ZZ(mu),0],
-		[0,0,0,0,0,ZZ(mu),0,ZZ(mu)]])
-
-	assert is_symplectic_matrix_dim4(M)
-
-	return M
-	
-
-# ========================================================== #
-#     Functions for the class KaniEndo (one isogeny chain)   #
-# ========================================================== #
-
-def gluing_base_change_matrix_dim2_dim4(a1,a2,m,mua2):
-	r"""Computes the symplectic base change matrix of a symplectic basis (*,B_K4) of Am*Am[4]
-	given by the kernel of the dimension 4 gluing isogeny Am*Am-->B:
-	
-	B_K4=2**(e-m)[(Phi([a1]P1,sigma(P1)),Phi([a2]P1,0)),(Phi([a1]P2,sigma(P2)),Phi([a2]P2,0)),
-	(Phi(-[a2]P1,0),Phi([a1]P1,sigma(P1))),(Phi(-[a2]P2,0),Phi([a2]P2,sigma(P2)))]
-	
-	in the basis associated to the product theta-structure of level 2 of Am*Am:
-	
-	B:=[(S1,0),(S2,0),(0,S1),(0,S2),(T1,0),(T2,0),(0,T1),(0,T2)]
-	
-	where:
-	- (P1,P2) is the canonical basis of E1[2**f].
-	- (R1,R2) is the image of (P1,P2) by sigma.
-	- Phi is the 2**m-isogeny E1*E2-->Am (m first steps of the chain in dimension 2).
-	- S1=[2**e]Phi([lamb]P2,[a]sigma(P1)+[b]sigma(P2)).
-	- S2=[2**e]Phi([mu]P1,[c]sigma(P1)+[d]sigma(P2)).
-	- T1=[2**(e-m)]Phi([a1]P1-[a2]P2,sigma(P1)).
-	- T2=[2**(e-m)]Phi([a1]P2+[a2]P1,sigma(P2)).
-	- (S1,S2,T1,T2) is induced by the image by Phi of a symplectic basis of E1*E2[2**(m+2)] lying
-	above the symplectic basis of E1*E2[4] outputted by gluing_base_change_matrix_dim2.
-
-	INPUT:
-	- a1, a2: integers.
-	- m: integer (number of steps in dimension 2).
-	- mua2: product mu*a2.
-
-	OUTPUT:
-	- M: symplectic base change matrix of (*,B_K4) in B.
-	"""
-	a1a2_2m=ZZ(a1*a2//2**m)
-	a22_2m=ZZ(a2**2//2**m)
-
-	Z4=Integers(4)
-
-	C=matrix(Z4,[[-a1a2_2m,a22_2m,a22_2m,a1a2_2m],
-		[-a22_2m,-a1a2_2m,-a1a2_2m,a22_2m],
-		[-a22_2m,-a1a2_2m,-a1a2_2m,a22_2m],
-		[a1a2_2m,-a22_2m,-a22_2m,-a1a2_2m]])
-
-	D=matrix(Z4,[[1,0,0,0],
-		[mua2,1,0,-mua2],
-		[0,0,1,0],
-		[0,mua2,mua2,1]])
-
-	M=complete_symplectic_matrix_dim4(C,D,4)
-
-	assert is_symplectic_matrix_dim4(M)
-
-	return M
-
-def splitting_base_change_matrix_dim4(a1,a2,q,m,M0,A_B,mu=None):
-	r"""
-	Let F be the endomorphism of E1^2*E2^2 given by Kani's lemma. Write: 
-			E1^2*E2^2 -- Phi x Phi --> Am^2 -- G --> E1^2*E2^2,
-	where Phi: E1 x E1 --> Am is a 2**m-isogeny in dimension 2. 
-	Let (U_1,...,U_4,V_1,...,V_4) be a symplectic basis of Am^2[2**(e-m+2)] 
-	such that V_i=Phi x Phi(W_i), where W_1,...,W_4 have order 2**(e+2), lie over ker(F) 
-	and generate an isotropic subgroup:
-	W_1=([a1]P1-[2^e/a1]P1,[a2]P1,R2,0)
-	W_2=([a1]Q1,[a2]Q1,S2,0)
-	W_3=(-[a2]P1,[a1]P1,0,R2)
-	W_4=(-[a2]Q1,[a1]Q1,0,S2),
-	with (P1,Q1), a basis of E1[2**(e+2)] and (R2,S2) its image via 
-	sigma: E1 --> E2. Then B:=([2^(e-m)]G(U_1),...,[2^(e-m)]G(U_4),G(V_1),...,G(V_4))
-	is a symplectic basis of E1^2*E2^2[4].
-
-	We assume that ([2^(e-m)]U_1,...,[2^(e-m)]U_4) is the symplectic complement of 
-	([2^(e-m)]V_1,...,[2^(e-m)]V_4) that has been outputted by 
-	gluing_base_change_matrix_dim2_dim4 for the gluing isogeny on Am^2 
-	(first 2-isogeny of G). This function computes the base change matrix of B
-	in the symplectic basis of E1^2*E2^2[4]:
-	B0=[(T1,0,0,0),(0,T1,0,0),(0,0,T2,0),(0,0,0,T2),(U1,0,0,0),(0,U1,0,0),
-	(0,0,U2,0),(0,0,0,U2)]
-	associated to the product Theta structure on E1^2*E2^2.
-
-	INPUT:
-	- a1,a2,q: integers defining F (q=deg(sigma)).
-	- m: 2-adic valuation of a2.
-	- M0: base change matrix of the symplectic basis 2**e*B1 of E1*E2[4] 
-	given by:
-	B1:=[[(P1,0),(0,R2)],[(Q1,0),(0,lamb*S2)]]
-	in the canonical symplectic basis:
-	B0:=[[(T1,0),(0,T2)],[(U1,0),(0,U2)]],
-	where lamb is the modular inverse of q mod 2**(e+2), so that:
-	e_{2**(e+2)}(P1,P2)=e_{2**(e+2)}(R1,lamb*R2).
-	- A_B: 4 first columns (left part) of the symplectic matrix outputted by
-	gluing_base_change_matrix_dim2_dim4.
-	- mu, a, b, c, d: integers defining the product Theta structure of Am^2
-	given by the four torsion basis [2**(e-m)]*B1 of Am, where:
-	B1=[[2**m]Phi([2**(m+1)]P2,[a]sigma(P1)+[b]sigma(P2)),
-	[2**m]Phi([mu]P1,[2**(m+1)]sigma(P1)+[d]sigma(P2)),
-	Phi([a1]P1-[a2]P2,sigma(P1)),
-	Phi([a1]P2+[a2]P1,sigma(P2))].
-	Only mu is given.
-
-	OUTPUT: The desired base change matrix.
-	"""
-	Z4=Integers(4)
-
-	a2_2m=ZZ(a2//2**m)
-	a12_q_2m=ZZ((a1**2+q)//2**m)
-
-	inv_q=inverse_mod(q,4)
-	inv_a1=inverse_mod(a1,4)
-
-	lamb=ZZ(2**(m+1))
-	if mu==None:
-		mu=ZZ((1-2**(m+1)*q)*inv_a1)
-	a=ZZ(2**(m+1)*a2*inv_q)
-	b=ZZ(-(1+2**(m+1)*a1)*inv_q)
-	c=ZZ(2**(m+1))
-	d=ZZ(-mu*a2*inv_q)
-
-	# Matrix of the four torsion basis of E1^2*E2^2[4] given by
-	# ([2^(e-m)]G(B1[0],0),[2^(e-m)]G(B1[1],0),[2^(e-m)]G(0,B1[0]),[2^(e-m)]G(0,B1[1]),
-	# G(B1[2],0),G(B1[3],0),G(0,B1[2]),G(0,B1[3])) in the basis induced by 
-	# [2**e](P1,Q1,R2,[1/q]S2)
-	M1=matrix(Z4,[[a*q,mu*a1+c*q,0,mu*a2,a12_q_2m,a1*a2_2m,a1*a2_2m,a2*a2_2m],
-		[0,-mu*a2,a*q,mu*a1+c*q,-a1*a2_2m,-a2*a2_2m,a12_q_2m,a1*a2_2m],
-		[a1*a,a1*c-mu,-a*a2,-c*a2,0,-a2_2m,-a2_2m,0],
-		[a2*a,a2*c,a*a1,c*a1-mu,a2_2m,0,0,-a2_2m],
-		[lamb*a1+b*q,d*q,lamb*a2,0,-a1*a2_2m,a12_q_2m,-a2*a2_2m,a1*a2_2m],
-		[-lamb*a2,0,lamb*a1+b*q,d*q,a2*a2_2m,-a1*a2_2m,-a1*a2_2m,a12_q_2m],
-		[(a1*b-lamb)*q,a1*d*q,-b*a2*q,-a2*d*q,a2_2m*q,0,0,-a2_2m*q],
-		[a2*b*q,a2*d*q,(b*a1-lamb)*q,a1*d*q,0,a2_2m*q,a2_2m*q,0]])
-	#A,B,C,D=bloc_decomposition(M1)
-	#if B.transpose()*A!=A.transpose()*B:
-		#print("B^T*A!=A^T*B")
-	#if C.transpose()*D!=D.transpose()*C:
-		#print("C^T*D!=D^T*C")
-	#if A.transpose()*D-B.transpose()*C!=identity_matrix(4):
-		#print(A.transpose()*D-B.transpose()*C)
-		#print("A^T*D-B^T*C!=I")
-	#print(M1)
-	#print(M1.det())
-
-	# Matrix of ([2^e]G(U_1),...,[2^e]G(U_4)) in the basis induced by
-	# [2**e](P1,Q1,R2,[1/q]S2)
-	M_left=M1*A_B
-	#print(A_B)
-	#print(M_left)
-
-	# Matrix of (G(V_1),...,G(V_4)) in the basis induced by [2**e](P1,Q1,R2,[1/q]S2)
-	M_right=matrix(Z4,[[0,0,0,0],
-		[a2*inv_a1,0,1,0],
-		[inv_a1,0,0,0],
-		[0,0,0,0],
-		[0,1,0,-a2*inv_a1],
-		[0,0,0,0],
-		[0,0,0,0],
-		[0,0,0,q*inv_a1]])
-
-	# Matrix of the basis induced by [2**e](P1,Q1,R2,[1/q]S2) in the basis 
-	# B0 (induced by T1, U1, T2, U2)
-	MM0=matrix(Z4,[[M0[0,0],0,M0[0,1],0,M0[0,2],0,M0[0,3],0],
-				  [0,M0[0,0],0,M0[0,1],0,M0[0,2],0,M0[0,3]],
-				  [M0[1,0],0,M0[1,1],0,M0[1,2],0,M0[1,3],0],
-				  [0,M0[1,0],0,M0[1,1],0,M0[1,2],0,M0[1,3]],
-				  [M0[2,0],0,M0[2,1],0,M0[2,2],0,M0[2,3],0],
-				  [0,M0[2,0],0,M0[2,1],0,M0[2,2],0,M0[2,3]],
-				  [M0[3,0],0,M0[3,1],0,M0[3,2],0,M0[3,3],0],
-				  [0,M0[3,0],0,M0[3,1],0,M0[3,2],0,M0[3,3]]])
-
-	M=MM0*block_matrix(1,2,[M_left,M_right])
-
-	#A,B,C,D=bloc_decomposition(M)
-
-	#M=complete_symplectic_matrix_dim4(C,D)
-
-	#print(M.det())
-	#print(M)
-
-	A,B,C,D=bloc_decomposition(M)
-	if B.transpose()*A!=A.transpose()*B:
-		print("B^T*A!=A^T*B")
-	if C.transpose()*D!=D.transpose()*C:
-		print("C^T*D!=D^T*C")
-	if A.transpose()*D-B.transpose()*C!=identity_matrix(4):
-		print("A^T*D-B^T*C!=I")
-	assert is_symplectic_matrix_dim4(M)
-
-	return M
-
-# ============================================================================ #
-#     Functions for the class KaniEndoHalf (isogeny chain decomposed in two)   #
-# ============================================================================ #
-
-def complete_kernel_matrix_F1(a1,a2,q,f):
-	r"""Computes the symplectic base change matrix of a symplectic basis of the form (*,B_Kp1) 
-	in the symplectic basis of E1^2*E2^2[2**f] given by:
-	B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R1,0),(0,0,0,R1)],
-	[(P2,0,0,0),(0,P2,0,0),(0,0,lamb*R2,0),(0,0,0,lamb*R2)]]
-	where:
-	- B_Kp1 is a basis of an isotropic subgroup of E1^2*E2^2[2**f] lying above ker(F1).
-	By convention B_Kp1=[(\tilde{\alpha}_1(P1,0),\Sigma(P1,0)),
-	(\tilde{\alpha}_1(P2,0),\Sigma(P2,0)),
-	(\tilde{\alpha}_1(0,P1),\Sigma(0,P1)),
-	(\tilde{\alpha}_1(0,P2),\Sigma(0,P2))]
-	- lamb is the inverse of q modulo 2**f.
-	- (P1,P2) is the canonical basis of E1[2**f].
-	- (R1,R2) is the image of (P1,P2) by sigma.
-
-	Input:
-	- a1, a2, q: Integers such that q+a1**2+a2**2=2**e.
-	- f: integer determining the accessible 2-torsion in E1 (E1[2**f]).
-
-	Output:
-	- M: symplectic base change matrix of (*,B_Kp1) in B1.
-	"""
-	N=2**f
-	ZN=Integers(N)
-
-	C=matrix(ZN,[[a1,0,-a2,0],
-		[a2,0,a1,0],
-		[1,0,0,0],
-		[0,0,1,0]])
-
-	D=matrix(ZN,[[0,a1,0,-a2],
-		[0,a2,0,a1],
-		[0,q,0,0],
-		[0,0,0,q]])
-
-	assert C.transpose()*D==D.transpose()*C
-
-	M=complete_symplectic_matrix_dim4(C,D,N)
-
-	assert is_symplectic_matrix_dim4(M)
-
-	return M
-
-def complete_kernel_matrix_F2_dual(a1,a2,q,f):
-	r"""Computes the symplectic base change matrix of a symplectic basis of the form (*,B_Kp2) 
-	in the symplectic basis of E1^2*E2^2[2**f] given by:
-	B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R1,0),(0,0,0,R1)],
-	[(P2,0,0,0),(0,P2,0,0),(0,0,lamb*R2,0),(0,0,0,lamb*R2)]]
-	where:
-	- B_Kp2 is a basis of an isotropic subgroup of E1^2*E2^2[2**f] lying above ker(\tilde{F2}).
-	By convention B_Kp2=[(\alpha_1(P1,0),-\Sigma(P1,0)),
-	(\alpha_1(P2,0),-\Sigma(P2,0)),
-	(\alpha_1(0,P1),-\Sigma(0,P1)),
-	(\alpha_1(0,P2),-\Sigma(0,P2))]. 
-	- lamb is the inverse of q modulo 2**f.
-	- (P1,P2) is the canonical basis of E1[2**f].
-	- (R1,R2) is the image of (P1,P2) by sigma.
-
-	Input:
-	- a1, a2, q: Integers such that q+a1**2+a2**2=2**e.
-	- f: integer determining the accessible 2-torsion in E1 (E1[2**f]).
-
-	Output:
-	- M: symplectic base change matrix of (*,B_Kp2) in B1.
-	"""
-	N=2**f
-	ZN=Integers(N)
-
-	C=matrix(ZN,[[a1,0,a2,0],
-		[-a2,0,a1,0],
-		[-1,0,0,0],
-		[0,0,-1,0]])
-
-	D=matrix(ZN,[[0,a1,0,a2],
-		[0,-a2,0,a1],
-		[0,-q,0,0],
-		[0,0,0,-q]])
-
-
-
-	M=complete_symplectic_matrix_dim4(C,D,N)
-
-	assert is_symplectic_matrix_dim4(M)
-
-	return M
-
-def matrix_F_dual(a1,a2,q,f):
-	r""" Computes the matrix of \tilde{F}(B1) in B1, where:
-	B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R1,0),(0,0,0,R1)],
-	[(P2,0,0,0),(0,P2,0,0),(0,0,lamb*R2,0),(0,0,0,lamb*R2)]]
-	as defined in complete_kernel_matrix_F2_dual.
-
-	Input:
-	- a1, a2, q: Integers such that q+a1**2+a2**2=2**e.
-	- f: integer determining the accessible 2-torsion in E1 (E1[2**f]).
-
-	Output:
-	- M: symplectic base change matrix of \tilde{F}(B1) in B1.
-	"""
-	N=2**f
-	ZN=Integers(N)
-
-	M=matrix(ZN,[[a1,-a2,-q,0,0,0,0,0],
-		[a2,a1,0,-q,0,0,0,0],
-		[1,0,a1,a2,0,0,0,0],
-		[0,1,-a2,a1,0,0,0,0],
-		[0,0,0,0,a1,-a2,-1,0],
-		[0,0,0,0,a2,a1,0,-1],
-		[0,0,0,0,q,0,a1,a2],
-		[0,0,0,0,0,q,-a2,a1]])
-
-	return M
-
-def matrix_F(a1,a2,q,f):
-	r""" Computes the matrix of F(B1) in B1, where:
-	B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R1,0),(0,0,0,R1)],
-	[(P2,0,0,0),(0,P2,0,0),(0,0,lamb*R2,0),(0,0,0,lamb*R2)]]
-	as defined in complete_kernel_matrix_F1.
-
-	Input:
-	- a1, a2, q: Integers such that q+a1**2+a2**2=2**e.
-	- f: integer determining the accessible 2-torsion in E1 (E1[2**f]).
-
-	Output:
-	- M: symplectic base change matrix of \tilde{F}(B1) in B1.
-	"""
-	N=2**f
-	ZN=Integers(N)
-
-	M=matrix(ZN,[[a1,a2,q,0,0,0,0,0],
-		[-a2,a1,0,q,0,0,0,0],
-		[-1,0,a1,-a2,0,0,0,0],
-		[0,-1,a2,a1,0,0,0,0],
-		[0,0,0,0,a1,a2,1,0],
-		[0,0,0,0,-a2,a1,0,1],
-		[0,0,0,0,-q,0,a1,-a2],
-		[0,0,0,0,0,-q,a2,a1]])
-
-	return M
-
-def starting_two_symplectic_matrices(a1,a2,q,f):
-	r"""
-	Computes the matrices of two symplectic basis of E1^2*E2^2[2**f] given 
-	by (*,B_Kp1) and (*,B_Kp2) in the basis
-	B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R1,0),(0,0,0,R1)],
-	[(P2,0,0,0),(0,P2,0,0),(0,0,lamb*R2,0),(0,0,0,lamb*R2)]]
-	as defined in complete_kernel_matrix_F1.
-
-	Input:
-	- a1, a2, q: Integers such that q+a1**2+a2**2=2**e.
-	- f: integer determining the accessible 2-torsion in E1 (E1[2**f]).
-
-	Output:
-	- M1, M2: the symplectic base change matrices of (*,B_Kp1) and (*,B_Kp2) in B1.
-	"""
-	M1_0=complete_kernel_matrix_F1(a1,a2,q,f)
-	MatF=matrix_F(a1,a2,q,f)
-
-	# Matrix of an isotropic subgroup of E1^2*E2^2[2**f] lying above ker(\tilde{F2}).
-	Block_right2=MatF*M1_0[:,[0,1,2,3]]
-
-	N=ZZ(2**f)
-
-	C=Block_right2[[0,1,2,3],:]
-	D=Block_right2[[4,5,6,7],:]
-
-	assert C.transpose()*D==D.transpose()*C
-
-	# Matrix of the resulting symplectic basis (*,B_Kp2)
-	M2=complete_symplectic_matrix_dim4(C,D,N)
-
-	MatF_dual=matrix_F_dual(a1,a2,q,f)
-
-	Block_right1=MatF_dual*M2[:,[0,1,2,3]]
-
-	C=Block_right1[[0,1,2,3],:]
-	D=Block_right1[[4,5,6,7],:]
-
-	A=M1_0[[0,1,2,3],[0,1,2,3]]
-	B=M1_0[[4,5,6,7],[0,1,2,3]]
-
-	assert C.transpose()*D==D.transpose()*C
-	assert B.transpose()*A==A.transpose()*B
-
-	# Matrix of the resulting symplectic basis (*,B_Kp1)
-	M1=block_matrix(1,2,[M1_0[:,[0,1,2,3]],-Block_right1])
-
-	assert is_symplectic_matrix_dim4(M1)
-
-	A,B,C,D=bloc_decomposition(M1)
-	a2_div=a2
-	m=0
-	while a2_div%2==0:
-		m+=1
-		a2_div=a2_div//2
-	for j in range(4):
-		assert (-D[0,j]*a1-C[0,j]*a2-D[2,j])%2**m==0
-		assert (C[0,j]*a1-D[0,j]*a2+C[2,j]*q)%2**m==0
-		assert (-D[1,j]*a1-C[1,j]*a2-D[3,j])%2**m==0
-		assert (C[1,j]*a1-D[1,j]*a2+C[3,j]*q)%2**m==0
-
-	return M1, M2
-
-def gluing_base_change_matrix_dim2_F1(a1,a2,q):
-	r"""Computes the symplectic base change matrix of a symplectic basis (*,B_K4) of E1*E2[4]
-	given by the kernel of the dimension 2 gluing isogeny:
-	B_K4=2**(f-2)[([a1]P1-[a2]P2,R1),([a1]P2+[a2]P1,R2)]
-	in the basis $2**(f-2)*B1$ given by:
-	B1:=[[(P1,0),(0,R1)],[(P2,0),(0,[1/q]*R2)]]
-	where:
-	- lamb is the inverse of q modulo 2**f.
-	- (P1,P2) is the canonical basis of E1[2**f].
-	- (R1,R2) is the image of (P1,P2) by sigma.
-
-	Input:
-	- a1, q: integers.
-
-	Output:
-	- M: symplectic base change matrix of (*,B_K4) in 2**(f-2)*B1.
-	"""
-	return gluing_base_change_matrix_dim2(a1,a2,q)
-
-def gluing_base_change_matrix_dim2_dim4_F1(a1,a2,q,m,M1):
-	r"""Computes the symplectic base change matrix of the symplectic basis Bp of Am*Am[4] induced
-	by the image of the symplectic basis (x_1, ..., x_4, y_1, ..., y_4) of E1^2*E2^2[2**(e1+2)]
-	adapted to ker(F1)=[4]<y_1, ..., y_4> in the basis associated to the product theta-structure 
-	of level 2 of Am*Am:
-	
-	B:=[(S1,0),(S2,0),(0,S1),(0,S2),(T1,0),(T2,0),(0,T1),(0,T2)]
-	
-	where:
-	- (P1,Q1) is the canonical basis of E1[2**f].
-	- (R2,S2) is the image of (P1,P2) by sigma.
-	- Phi is the 2**m-isogeny E1*E2-->Am (m first steps of the chain in dimension 2).
-	- S1=[2**e]Phi([lamb]Q1,[a]sigma(P1)+[b]sigma(Q1)).
-	- S2=[2**e]Phi([mu]P1,[c]sigma(P1)+[d]sigma(Q1)).
-	- T1=[2**(e-m)]Phi([a1]P1-[a2]Q1,sigma(P1)).
-	- T2=[2**(e-m)]Phi([a1]Q1+[a2]P1,sigma(Q1)).
-	- (S1,S2,T1,T2) is induced by the image by Phi of a symplectic basis of E1*E2[2**(m+2)] lying
-	above the symplectic basis of E1*E2[4] outputted by gluing_base_change_matrix_dim2.
-
-	INPUT:
-	- a1, a2, q: integers.
-	- m: integer (number of steps in dimension 2 and 2-adic valuation of a2).
-	- M1: matrix of (x_1, ..., x_4, y_1, ..., y_4) in the symplectic basis of E1^2*E2^2[2**(e1+2)]
-	given by:
-
-	B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R2,0),(0,0,0,R2)],
-	[(Q1,0,0,0),(0,Q1,0,0),(0,0,[1/q]*S2,0),(0,0,0,[1/q]*S2)]]
-
-	OUTPUT:
-	- M: symplectic base change matrix of Bp in B.
-	"""
-
-	inv_a1=inverse_mod(a1,2**(m+2))
-	inv_q=inverse_mod(q,2**(m+2))
-	lamb=ZZ(2**(m+1))
-	mu=ZZ((1-2**(m+1)*q)*inv_a1)
-	a=ZZ(2**(m+1)*a2*inv_q)
-	bq=ZZ((-1-2**(m+1)*a1))
-	c=ZZ(2**(m+1))
-	dq=-ZZ(mu*a2)
-
-	Z4=Integers(4)
-
-	A,B,C,D=bloc_decomposition(M1)
-
-	Ap=matrix(Z4,[[ZZ(-B[0,j]*a1-A[0,j]*a2-B[2,j]) for j in range(4)],
-		[ZZ(A[0,j]*a1-B[0,j]*a2+A[2,j]*q) for j in range(4)],
-		[ZZ(-B[1,j]*a1-A[1,j]*a2-B[3,j]) for j in range(4)],
-		[ZZ(A[1,j]*a1-B[1,j]*a2+A[3,j]*q) for j in range(4)]])
-
-	Bp=matrix(Z4,[[ZZ(2**m*(B[2,j]*a-A[0,j]*lamb-A[2,j]*bq)) for j in range(4)],
-		[ZZ(2**m*(B[2,j]*c+B[0,j]*mu-A[2,j]*dq)) for j in range(4)],
-		[ZZ(2**m*(B[3,j]*a-A[1,j]*lamb-A[3,j]*bq)) for j in range(4)],
-		[ZZ(2**m*(B[3,j]*c+B[1,j]*mu-A[3,j]*dq)) for j in range(4)]])
-
-	Cp=matrix(Z4,[[ZZ(ZZ(-D[0,j]*a1-C[0,j]*a2-D[2,j])//(2**m)) for j in range(4)],
-		[ZZ(ZZ(C[0,j]*a1-D[0,j]*a2+C[2,j]*q)//2**m) for j in range(4)],
-		[ZZ(ZZ(-D[1,j]*a1-C[1,j]*a2-D[3,j])//2**m) for j in range(4)],
-		[ZZ(ZZ(C[1,j]*a1-D[1,j]*a2+C[3,j]*q)//2**m) for j in range(4)]])
-
-	Dp=matrix(Z4,[[ZZ(D[2,j]*a-C[0,j]*lamb-C[2,j]*bq) for j in range(4)],
-		[ZZ(D[0,j]*mu+D[2,j]*c-C[2,j]*dq) for j in range(4)],
-		[ZZ(D[3,j]*a-C[1,j]*lamb-C[3,j]*bq) for j in range(4)],
-		[ZZ(D[1,j]*mu+D[3,j]*c-C[3,j]*dq) for j in range(4)]])
-
-	M=block_matrix(2,2,[[Ap,Cp],[Bp,Dp]])
-
-	assert is_symplectic_matrix_dim4(M)
-
-	return M
-
-def gluing_base_change_matrix_dim2_F2(a1,a2,q):
-	r"""Computes the symplectic base change matrix of a symplectic basis (*,B_K4) of E1*E2[4]
-	given by the kernel of the dimension 2 gluing isogeny:
-	B_K4=2**(f-2)[([a1]P1-[a2]P2,R1),([a1]P2+[a2]P1,R2)]
-	in the basis $2**(f-2)*B1$ given by:
-	B1:=[[(P1,0),(0,R1)],[(P2,0),(0,[1/q]*R2)]]
-	where:
-	- lamb is the inverse of q modulo 2**f.
-	- (P1,P2) is the canonical basis of E1[2**f].
-	- (R1,R2) is the image of (P1,P2) by sigma.
-
-	Input:
-	- a1, q: integers.
-
-	Output:
-	- M: symplectic base change matrix of (*,B_K4) in 2**(f-2)*B1.
-	"""
-
-	Z4=Integers(4)
-
-	mu=inverse_mod(a1,4)
-
-	A=matrix(Z4,[[0,mu],
-		[0,0]])
-	B=matrix(Z4,[[0,0],
-		[1,-ZZ(mu*a2)]])
-
-	C=matrix(Z4,[[ZZ(a1),-ZZ(a2)],
-		[-1,0]])
-	D=matrix(Z4,[[ZZ(a2),ZZ(a1)],
-		[0,-ZZ(q)]])
-
-	M=block_matrix([[A,C],[B,D]])
-
-	assert is_symplectic_matrix_dim2(M)
-
-	return M
-
-def gluing_base_change_matrix_dim2_dim4_F2(a1,a2,q,m,M2):
-	r"""Computes the symplectic base change matrix of the symplectic basis Bp of Am*Am[4] induced
-	by the image of the symplectic basis (x_1, ..., x_4, y_1, ..., y_4) of E1^2*E2^2[2**(e1+2)]
-	adapted to ker(F1)=[4]<y_1, ..., y_4> in the basis associated to the product theta-structure 
-	of level 2 of Am*Am:
-	
-	B:=[(S1,0),(S2,0),(0,S1),(0,S2),(T1,0),(T2,0),(0,T1),(0,T2)]
-	
-	where:
-	- (P1,Q1) is the canonical basis of E1[2**f].
-	- (R2,S2) is the image of (P1,P2) by sigma.
-	- Phi is the 2**m-isogeny E1*E2-->Am (m first steps of the chain in dimension 2).
-	- S1=[2**e]Phi([lamb]Q1,[a]sigma(P1)+[b]sigma(Q1)).
-	- S2=[2**e]Phi([mu]P1,[c]sigma(P1)+[d]sigma(Q1)).
-	- T1=[2**(e-m)]Phi([a1]P1-[a2]Q1,sigma(P1)).
-	- T2=[2**(e-m)]Phi([a1]Q1+[a2]P1,sigma(Q1)).
-	- (S1,S2,T1,T2) is induced by the image by Phi of a symplectic basis of E1*E2[2**(m+2)] lying
-	above the symplectic basis of E1*E2[4] outputted by gluing_base_change_matrix_dim2.
-
-	INPUT:
-	- a1, a2, q: integers.
-	- m: integer (number of steps in dimension 2 and 2-adic valuation of a2).
-	- M2: matrix of (x_1, ..., x_4, y_1, ..., y_4) in the symplectic basis of E1^2*E2^2[2**(e1+2)]
-	given by:
-
-	B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R2,0),(0,0,0,R2)],
-	[(Q1,0,0,0),(0,Q1,0,0),(0,0,[1/q]*S2,0),(0,0,0,[1/q]*S2)]]
-
-	OUTPUT:
-	- M: symplectic base change matrix of Bp in B.
-	"""
-
-	inv_a1=inverse_mod(a1,2**(m+2))
-	inv_q=inverse_mod(q,2**(m+2))
-	lamb=ZZ(2**(m+1))
-	mu=ZZ((1+2**(m+1)*q)*inv_a1)
-	a=ZZ(2**(m+1)*a2*inv_q)
-	bq=ZZ((1+2**(m+1)*a1))
-	c=ZZ(2**(m+1))
-	dq=-ZZ(mu*a2)
-
-	Z4=Integers(4)
-
-	A,B,C,D=bloc_decomposition(M2)
-
-	Ap=matrix(Z4,[[ZZ(-B[0,j]*a1+A[0,j]*a2+B[2,j]) for j in range(4)],
-		[ZZ(A[0,j]*a1+B[0,j]*a2-A[2,j]*q) for j in range(4)],
-		[ZZ(-B[1,j]*a1+A[1,j]*a2+B[3,j]) for j in range(4)],
-		[ZZ(A[1,j]*a1+B[1,j]*a2-A[3,j]*q) for j in range(4)]])
-
-	Bp=matrix(Z4,[[ZZ(2**m*(B[2,j]*a-A[0,j]*lamb-A[2,j]*bq)) for j in range(4)],
-		[ZZ(2**m*(B[2,j]*c+B[0,j]*mu-A[2,j]*dq)) for j in range(4)],
-		[ZZ(2**m*(B[3,j]*a-A[1,j]*lamb-A[3,j]*bq)) for j in range(4)],
-		[ZZ(2**m*(B[3,j]*c+B[1,j]*mu-A[3,j]*dq)) for j in range(4)]])
-
-	Cp=matrix(Z4,[[ZZ(ZZ(-D[0,j]*a1+C[0,j]*a2+D[2,j])//(2**m)) for j in range(4)],
-		[ZZ(ZZ(C[0,j]*a1+D[0,j]*a2-C[2,j]*q)//2**m) for j in range(4)],
-		[ZZ(ZZ(-D[1,j]*a1+C[1,j]*a2+D[3,j])//2**m) for j in range(4)],
-		[ZZ(ZZ(C[1,j]*a1+D[1,j]*a2-C[3,j]*q)//2**m) for j in range(4)]])
-
-	Dp=matrix(Z4,[[ZZ(D[2,j]*a-C[0,j]*lamb-C[2,j]*bq) for j in range(4)],
-		[ZZ(D[0,j]*mu+D[2,j]*c-C[2,j]*dq) for j in range(4)],
-		[ZZ(D[3,j]*a-C[1,j]*lamb-C[3,j]*bq) for j in range(4)],
-		[ZZ(D[1,j]*mu+D[3,j]*c-C[3,j]*dq) for j in range(4)]])
-
-	M=block_matrix(2,2,[[Ap,Cp],[Bp,Dp]])
-
-	assert is_symplectic_matrix_dim4(M)
-
-	return M
-
-def point_matrix_product(M,L_P,J=None,modulus=None):
-	r"""
-	Input:
-	- M: matrix with (modular) integer values.
-	- L_P: list of elliptic curve points [P1,P2,R1,R2] such that the rows of M correspond to the vectors
-	(P1,0,0,0),(0,P1,0,0),(0,0,R1,0),(0,0,0,R1),(P2,0,0,0),(0,P2,0,0),(0,0,R2,0),(0,0,0,R2).
-	- J: list of column indices (default, all the columns).
-	- modulus: order of points in L_P (default, None).
-
-	Output:
-	- L_ret: list of points corresponding to the columns of M with indices in J.
-	"""
-	if modulus==None:
-		M1=M
-	else:
-		Zmod=Integers(modulus)
-		M1=matrix(Zmod,M)
-
-	if J==None:
-		J=range(M1.ncols())
-
-	L_ret=[]
-	for j in J:
-		L_ret.append(TuplePoint(M1[0,j]*L_P[0]+M1[4,j]*L_P[1],M1[1,j]*L_P[0]+M1[5,j]*L_P[1],
-			M1[2,j]*L_P[2]+M1[6,j]*L_P[3],M1[3,j]*L_P[2]+M1[7,j]*L_P[3]))
-
-	return L_ret
-
-
-def kernel_basis(M,ei,mP1,mP2,mR1,mlambR2):
-	r"""
-	Input:
-	- M: matrix of a symplectic basis in the basis
-	B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R1,0),(0,0,0,R1)],
-	[(P2,0,0,0),(0,P2,0,0),(0,0,lamb*R2,0),(0,0,0,lamb*R2)]]
-	as defined in complete_kernel_matrix_F1.
-	- ei: length of F1 or F2.
-	- mP1,mP2: canonical basis (P1,P2) of E1[2**f] multiplied by m:=2**(f-ei-2).
-	- mR1,mlambR2: (mR1,mlambR2)=(m*sigma(P1),m*sigma(P2)), where lamb is the
-	inverse of q=deg(sigma) modulo 2**f.
-
-	Output:
-	- Basis of the second symplectic subgroup basis of E1^2*E2^2[2**(ei+2)] induced by M. 
-	"""
-	modulus=2**(ei+2)
-
-	return point_matrix_product(M,[mP1,mP2,mR1,mlambR2],[4,5,6,7],modulus)
-
-def base_change_canonical_dim4(P1,P2,R1,R2,q,f,e1,e2):
-	lamb=inverse_mod(q,2**f)
-	
-	lambR2=lamb*R2
-
-	P1_doubles=[P1]
-	P2_doubles=[P2]
-	R1_doubles=[R1]
-	lambR2_doubles=[lambR2]
-
-	for i in range(f-2):
-		P1_doubles.append(2*P1_doubles[-1])
-		P2_doubles.append(2*P2_doubles[-1])
-		R1_doubles.append(2*R1_doubles[-1])
-		lambR2_doubles.append(2*lambR2_doubles[-1])
-
-	# Constructing canonical basis of E1[4] and E2[4].
-	_,_,T1,T2,MT=make_canonical(P1_doubles[-1],P2_doubles[-1],4,preserve_pairing=True)
-	_,_,U1,U2,MU=make_canonical(R1_doubles[-1],lambR2_doubles[-1],4,preserve_pairing=True)
-
-	# Base change matrix of the symplectic basis 2**(f-2)*B1 of E1^2*E2^2[4] in the basis:
-	# B0:=[[(T1,0,0,0),(0,T1,0,0),(0,0,U1,0),(0,0,0,U1)],
-	#[(T2,0,0,0),(0,T2,0,0),(0,0,U2,0),(0,0,0,U2)]]
-	# where B1:=[[(P1,0,0,0),(0,P1,0,0),(0,0,R1,0),(0,0,0,R1)],
-	#[(P2,0,0,0),(0,P2,0,0),(0,0,lamb*R2,0),(0,0,0,lamb*R2)]]
-	Z4=Integers(4)
-	M0=matrix(Z4,[[MT[0,0],0,0,0,MT[1,0],0,0,0],
-		[0,MT[0,0],0,0,0,MT[1,0],0,0],
-		[0,0,MU[0,0],0,0,0,MU[1,0],0],
-		[0,0,0,MU[0,0],0,0,0,MU[1,0]],
-		[MT[0,1],0,0,0,MT[1,1],0,0,0],
-		[0,MT[0,1],0,0,0,MT[1,1],0,0],
-		[0,0,MU[0,1],0,0,0,MU[1,1],0],
-		[0,0,0,MU[0,1],0,0,0,MU[1,1]]])
-
-	return P1_doubles,P2_doubles,R1_doubles,lambR2_doubles,T1,T2,U1,U2,MT,MU,M0