DeRhamComputation/superelliptic.ipynb

def basis_holomorphic_differentials_degree(f, m, p):
    r = f.degree()
    delta = GCD(r, m)
    Rx.<x> = PolynomialRing(GF(p))
    Rxy.<x, y> = PolynomialRing(GF(p), 2)
    Fxy = FractionField(Rxy)
    #########basis of holomorphic differentials and de Rham
                
    basis_holo = []
    degrees0 = {}
    k = 0
        
    for j in range(1, m):
        for i in range(1, r):
            if (r*j - m*i >= delta):
                basis_holo += [Fxy(x^(i-1)/y^j)]
                degrees0[k] = (i-1, j)
                k = k+1
                    
    return(basis_holo, degrees0)

def holomorphic_differentials_basis(f, m, p):
        basis_holo, degrees0 = basis_holomorphic_differentials_degree(f, m, p)
        return basis_holo
        
def degrees_holomorphic_differentials(f, m, p):
    basis_holo, degrees0 = basis_holomorphic_differentials_degree(f, m, p)
    return degrees0
    
def basis_de_rham_degrees(f, m, p):
    r = f.degree()
    delta = GCD(r, m)
    Rx.<x> = PolynomialRing(GF(p))
    Rxy.<x, y> = PolynomialRing(GF(p), 2)
    Fxy = FractionField(Rxy)
    basis_holo = holomorphic_differentials_basis(f, m, p)
    basis = []
    for k in range(0, len(basis_holo)):
        basis += [(basis_holo[k], Rx(0))]

    ## non-holomorphic elts of H^1_dR
    t = len(basis)
    degrees0 = {}
    degrees1 = {}
    for j in range(1, m):
        for i in range(1, r):
            if (r*(m-j) - m*i >= delta): 
                s = Rx(m-j)*Rx(x)*Rx(f.derivative()) - Rx(m)*Rx(i)*f
                psi = Rx(cut(s, i))
                basis += [(Fxy(psi/y^j), Fxy(m*y^(m-j)/x^i))]
                degrees0[t] = (psi.degree(), j)
                degrees1[t] = (-i, m-j)
                t += 1
    return basis, degrees0, degrees1

def de_rham_basis(f, m, p):
    basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)
    return basis

def degrees_de_rham0(f, m, p):
    basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)
    return degrees0

def degrees_de_rham1(f, m, p):
    basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)
    return degrees1    


class superelliptic:
    
    def __init__(self, f, m, p):
        Rx.<x> = PolynomialRing(GF(p))
        Rxy.<x, y> = PolynomialRing(GF(p), 2)
        Fxy = FractionField(Rxy)
        self.polynomial = Rx(f)
        self.exponent = m
        self.characteristic = p
        
        r = Rx(f).degree()
        delta = GCD(r, m)
        self.degree_holo = degrees_holomorphic_differentials(f, m, p)
        self.degree_de_rham0 = degrees_de_rham0(f, m, p)
        self.degree_de_rham1 = degrees_de_rham1(f, m, p)
        
        holo_basis = holomorphic_differentials_basis(f, m, p)
        holo_basis_converted = []
        for a in holo_basis:
            holo_basis_converted += [superelliptic_form(self, a)]
        
        self.basis_holomorphic_differentials = holo_basis_converted
        

        dr_basis = de_rham_basis(f, m, p)
        dr_basis_converted = []
        for (a, b) in dr_basis:
            dr_basis_converted += [superelliptic_cech(self, superelliptic_form(self, a), superelliptic_function(self, b))]
        
        self.basis_de_rham = dr_basis_converted
        
    def __repr__(self):
        f = self.polynomial
        m = self.exponent
        p = self.characteristic
        return 'Superelliptic curve with the equation y^' + str(m) + ' = ' + str(f)+' over finite field with ' + str(p) + ' elements.'
    
    def is_smooth(self):
        f = self.polynomial
        if f.discriminant() == 0:
            return 0
        return 1
    
    def genus(self):
        r = self.polynomial.degree()
        m = self.exponent
        delta = GCD(r, m)
        return 1/2*((r-1)*(m-1) - delta + 1)
    
    def verschiebung_matrix(self):
        basis = self.basis_de_rham
        g = self.genus()
        p = self.characteristic
        M = matrix(GF(p), 2*g, 2*g)
        for i in range(0, len(basis)):
            w = basis[i]
            v = w.verschiebung().coordinates()
            M[i, :] = v
        return M
    
    def frobenius_matrix(self):
        basis = self.basis_de_rham
        g = self.genus()
        p = self.characteristic
        M = matrix(GF(p), 2*g, 2*g)
        
        for i in range(0, len(basis)):
            w = basis[i]
            v = w.frobenius().coordinates()
            M[i, :] = v
        return M

    def cartier_matrix(self):
        basis = self.basis_holomorphic_differentials
        g = self.genus()
        p = self.characteristic
        M = matrix(GF(p), g, g)
        for i in range(0, len(basis)):
            w = basis[i]
            v = w.cartier().coordinates()
            M[i, :] = v
        return M     
    
    def p_rank(self):
        return self.cartier_matrix().rank()
    
    def final_type(self, test = 0):
        F = self.frobenius_matrix()
        V = self.verschiebung_matrix()
        p = self.characteristic
        return flag(F, V, p, test)
    
def reduction(C, g):
    p = C.characteristic
    Rxy.<x, y> = PolynomialRing(GF(p), 2)
    Fxy = FractionField(Rxy)
    f = C.polynomial
    r = f.degree()
    m = C.exponent
    g = Fxy(g)
    g1 = g.numerator()
    g2 = g.denominator()
    
    Rx.<x> = PolynomialRing(GF(p))
    Fx = FractionField(Rx)
    FxRy.<y> = PolynomialRing(Fx)    
    (A, B, C) = xgcd(FxRy(g2), FxRy(y^m - f))
    g = FxRy(g1*B/A)
    
    while(g.degree(Rxy(y)) >= m):
        d = g.degree(Rxy(y))
        G = coff(g, d)
        i = floor(d/m)
        g = g - G*y^d + f^i * y^(d%m) *G
    
    return(FxRy(g))

def reduction_form(C, g):
    p = C.characteristic
    Rxy.<x, y> = PolynomialRing(GF(p), 2)
    Fxy = FractionField(Rxy)
    f = C.polynomial
    r = f.degree()
    m = C.exponent
    g = reduction(C, g)

    g1 = Rxy(0)
    Rx.<x> = PolynomialRing(GF(p))
    Fx = FractionField(Rx)
    FxRy.<y> = PolynomialRing(Fx)
    
    g = FxRy(g)
    for j in range(0, m):
        if j==0:
            G = coff(g, 0)
            g1 += FxRy(G)
        else:
            G = coff(g, j)
            g1 += Fxy(y^(j-m)*f*G)
    return(g1)
        
class superelliptic_function:
    def __init__(self, C, g):
        p = C.characteristic
        Rxy.<x, y> = PolynomialRing(GF(p), 2)
        Fxy = FractionField(Rxy)
        f = C.polynomial
        r = f.degree()
        m = C.exponent
        
        self.curve = C
        g = reduction(C, g)
        self.function = g
        
    def __repr__(self):
        return str(self.function)
    
    def jth_component(self, j):
        g = self.function
        C = self.curve
        p = C.characteristic
        Rx.<x> = PolynomialRing(GF(p))
        Fx.<x> = FractionField(Rx)
        FxRy.<y> = PolynomialRing(Fx)
        g = FxRy(g)
        return coff(g, j)
    
    def __add__(self, other):
        C = self.curve
        g1 = self.function
        g2 = other.function
        g = reduction(C, g1 + g2)
        return superelliptic_function(C, g)
    
    def __sub__(self, other):
        C = self.curve
        g1 = self.function
        g2 = other.function
        g = reduction(C, g1 - g2)
        return superelliptic_function(C, g)
    
    def __mul__(self, other):
        C = self.curve
        g1 = self.function
        g2 = other.function
        g = reduction(C, g1 * g2)
        return superelliptic_function(C, g)
    
    def __truediv__(self, other):
        C = self.curve
        g1 = self.function
        g2 = other.function
        g = reduction(C, g1 / g2)
        return superelliptic_function(C, g)
    
def diffn(self):
    C = self.curve
    f = C.polynomial
    m = C.exponent
    p = C.characteristic
    g = self.function
    Rxy.<x, y> = PolynomialRing(GF(p), 2)
    Fxy = FractionField(Rxy)
    g = Fxy(g)
    A = g.derivative(x)
    B = g.derivative(y)*f.derivative(x)/(m*y^(m-1))
    return superelliptic_form(C, A+B)
        
class superelliptic_form:
    def __init__(self, C, g):
        p = C.characteristic
        Rxy.<x, y> = PolynomialRing(GF(p), 2)
        Fxy = FractionField(Rxy)
        g = Fxy(reduction_form(C, g))
        self.form = g
        self.curve = C      
        
    def __add__(self, other):
        C = self.curve
        g1 = self.form
        g2 = other.form
        g = reduction(C, g1 + g2)
        return superelliptic_form(C, g)
    
    def __sub__(self, other):
        C = self.curve
        g1 = self.form
        g2 = other.form
        g = reduction(C, g1 - g2)
        return superelliptic_form(C, g)
    
    def __repr__(self):
        g = self.form
        if len(str(g)) == 1:
            return str(g) + ' dx'
        return '('+str(g) + ') dx'

    def __rmul__(self, constant):
        C = self.curve
        omega = self.form
        return superelliptic_form(C, constant*omega)        
    
    def cartier(self):
        C = self.curve
        m = C.exponent
        p = C.characteristic
        f = C.polynomial
        Rx.<x> = PolynomialRing(GF(p))
        Fx = FractionField(Rx)
        FxRy.<y> = PolynomialRing(Fx)
        Fxy = FractionField(FxRy)
        result = superelliptic_form(C, FxRy(0))
        mult_order = Integers(m)(p).multiplicative_order()
        M = Integer((p^(mult_order)-1)/m)
        
        for j in range(1, m):
            fct_j = self.jth_component(j)
            h = Rx(fct_j*f^(M*j))
            j1 = (p^(mult_order-1)*j)%m
            B = floor(p^(mult_order-1)*j/m)
            result += superelliptic_form(C, polynomial_part(p, h)/(f^B*y^(j1)))
        return result   
    
    def coordinates(self):
        C = self.curve
        p = C.characteristic
        m = C.exponent
        Rx.<x> = PolynomialRing(GF(p))
        Fx = FractionField(Rx)
        FxRy.<y> = PolynomialRing(Fx)
        g = C.genus()
        degrees_holo = C.degree_holo
        degrees_holo_inv = {b:a for a, b in degrees_holo.items()}
        basis = C.basis_holomorphic_differentials
        
        for j in range(1, m):
            omega_j = Fx(self.jth_component(j))
            if omega_j != Fx(0):
                d = degree_of_rational_fctn(omega_j, p)
                index = degrees_holo_inv[(d, j)]
                a = coeff_of_rational_fctn(omega_j, p)
                a1 = coeff_of_rational_fctn(basis[index].jth_component(j), p)
                elt = self - (a/a1)*basis[index]
                return elt.coordinates() + a/a1*vector([GF(p)(i == index) for i in range(0, g)])
            
        return vector(g*[0])
    
    def jth_component(self, j):
        g = self.form
        C = self.curve
        p = C.characteristic
        Rx.<x> = PolynomialRing(GF(p))
        Fx = FractionField(Rx)
        FxRy.<y> = PolynomialRing(Fx)
        Fxy = FractionField(FxRy)
        Ryinv.<y_inv> = PolynomialRing(Fx)
        g = Fxy(g)
        g = g(y = 1/y_inv)
        g = Ryinv(g)
        return coff(g, j)
    
    def is_regular_on_U0(self):
        C = self.curve
        p = C.characteristic
        m = C.exponent
        Rx.<x> = PolynomialRing(GF(p))
        for j in range(1, m):
            if self.jth_component(j) not in Rx:
                return 0
        return 1
    
    def is_regular_on_Uinfty(self):
        C = self.curve
        p = C.characteristic
        m = C.exponent
        f = C.polynomial
        r = f.degree()
        delta = GCD(m, r)
        M = m/delta
        R = r/delta
        
        for j in range(1, m):
            A = self.jth_component(j)
            d = degree_of_rational_fctn(A, p)
            if(-d*M + j*R -(M+1)<0):
                return 0
        return 1
    
    
class superelliptic_cech:
    def __init__(self, C, omega, fct):
        self.omega0 = omega
        self.omega8 = omega - diffn(fct)
        self.f = fct
        self.curve = C
        
    def __add__(self, other):
        C = self.curve
        return superelliptic_cech(C, self.omega0 + other.omega0, self.f + other.f)
    
    def __sub__(self, other):
        C = self.curve
        return superelliptic_cech(C, self.omega0 - other.omega0, self.f - other.f)

    def __rmul__(self, constant):
        C = self.curve
        w1 = self.omega0.form
        f1 = self.f.function
        w2 = superelliptic_form(C, constant*w1)
        f2 = superelliptic_function(C, constant*f1)
        return superelliptic_cech(C, w2, f2)    
    
    def __repr__(self):
        return "(" + str(self.omega0) + ", " + str(self.f) + ", " + str(self.omega8) + ")" 
    
    def verschiebung(self):
        C = self.curve
        omega = self.omega0
        p = C.characteristic
        Rx.<x> = PolynomialRing(GF(p))
        return superelliptic_cech(C, omega.cartier(), superelliptic_function(C, Rx(0)))
    
    def frobenius(self):
        C = self.curve
        fct = self.f.function
        p = C.characteristic
        Rx.<x> = PolynomialRing(GF(p))
        return superelliptic_cech(C, superelliptic_form(C, Rx(0)), superelliptic_function(C, fct^p))

    def coordinates(self):
        C = self.curve
        p = C.characteristic
        m = C.exponent
        Rx.<x> = PolynomialRing(GF(p))
        Fx = FractionField(Rx)
        FxRy.<y> = PolynomialRing(Fx)
        g = C.genus()
        degrees_holo = C.degree_holo
        degrees_holo_inv = {b:a for a, b in degrees_holo.items()}
        degrees0 = C.degree_de_rham0
        degrees0_inv = {b:a for a, b in degrees0.items()}
        degrees1 = C.degree_de_rham1
        degrees1_inv = {b:a for a, b in degrees1.items()}
        basis = C.basis_de_rham
        
        omega = self.omega0
        fct = self.f
        
        if fct.function == Rx(0) and omega.form != Rx(0):
            for j in range(1, m):
                omega_j = Fx(omega.jth_component(j))
                if omega_j != Fx(0):
                    d = degree_of_rational_fctn(omega_j, p)
                    index = degrees_holo_inv[(d, j)]
                    a = coeff_of_rational_fctn(omega_j, p)
                    a1 = coeff_of_rational_fctn(basis[index].omega0.jth_component(j), p)
                    elt = self - (a/a1)*basis[index]
                    return elt.coordinates() + a/a1*vector([GF(p)(i == index) for i in range(0, 2*g)])
                    
        for j in range(1, m):
            fct_j = Fx(fct.jth_component(j))
            if (fct_j != Rx(0)):
                d = degree_of_rational_fctn(fct_j, p)
            
                if (d, j) in degrees1.values():
                    index = degrees1_inv[(d, j)]
                    a = coeff_of_rational_fctn(fct_j, p)
                    elt = self - (a/m)*basis[index]
                    return elt.coordinates() + a/m*vector([GF(p)(i == index) for i in range(0, 2*g)])
                
                if d<0:
                    a = coeff_of_rational_fctn(fct_j, p)
                    h = superelliptic_function(C, FxRy(a*y^j*x^d))
                    elt = superelliptic_cech(C, self.omega0, self.f - h)
                    return elt.coordinates()
            
                if (fct_j != Rx(0)):
                    G = superelliptic_function(C, y^j*x^d)
                    a = coeff_of_rational_fctn(fct_j, p)
                    elt =self - a*superelliptic_cech(C, diffn(G), G)
                    return elt.coordinates()

        return vector(2*g*[0])
    
    def is_cocycle(self):
        w0 = self.omega0
        w8 = self.omega8
        fct = self.f
        if not w0.is_regular_on_U0() and not w8.is_regular_on_Uinfty():
            return('w0 & w8')
        if not w0.is_regular_on_U0():
            return('w0')
        if not w8.is_regular_on_Uinfty():
            return('w8')
        if w0.is_regular_on_U0() and w8.is_regular_on_Uinfty():
            return 1
        return 0
        
def degree_of_rational_fctn(f, p):
    Rx.<x> = PolynomialRing(GF(p))
    Fx = FractionField(Rx)
    f = Fx(f)
    f1 = f.numerator()
    f2 = f.denominator()
    d1 = f1.degree()
    d2 = f2.degree()
    return(d1 - d2)

def coeff_of_rational_fctn(f, p):
    Rx.<x> = PolynomialRing(GF(p))
    Fx = FractionField(Rx)
    f = Fx(f)
    if f == Rx(0):
        return 0
    f1 = f.numerator()
    f2 = f.denominator()
    d1 = f1.degree()
    d2 = f2.degree()
    a1 = f1.coefficients(sparse = false)[d1]
    a2 = f2.coefficients(sparse = false)[d2]
    return(a1/a2)

def coff(f, d):
    lista = f.coefficients(sparse = false)
    if len(lista) <= d:
        return 0
    return lista[d]

def cut(f, i):
    R = f.parent()
    coeff = f.coefficients(sparse = false)
    return sum(R(x^(j-i-1)) * coeff[j] for j in range(i+1, f.degree() + 1))

def polynomial_part(p, h):
    Rx.<x> = PolynomialRing(GF(p))
    h = Rx(h)
    result = Rx(0)
    for i in range(0, h.degree()+1):
        if (i%p) == p-1:
            power = Integer((i-(p-1))/p)
            result += Integer(h[i]) * x^(power)    
    return result

def preimage(U, V, M): #preimage of subspace U under M
    basis_preimage = M.right_kernel().basis()
    imageU = U.intersection(M.transpose().image())
    basis = imageU.basis()
    for v in basis:
        w = M.solve_right(v)
        basis_preimage = basis_preimage + [w]
    return V.subspace(basis_preimage)

def image(U, V, M):
    basis = U.basis()
    basis_image = []
    for v in basis:
        basis_image += [M*v]
    return V.subspace(basis_image)

def flag(F, V, p, test = 0):
    dim = F.dimensions()[0]
    space = VectorSpace(GF(p), dim)
    flag_subspaces = (dim+1)*[0]
    flag_used = (dim+1)*[0]
    final_type = (dim+1)*['?']
    
    flag_subspaces[dim] = space
    flag_used[dim] = 1
   
    
    while 1 in flag_used:
        index = flag_used.index(1)
        flag_used[index] = 0
        U = flag_subspaces[index]
        U_im = image(U, space, V)
        d_im = U_im.dimension()
        final_type[index] = d_im
        U_pre = preimage(U, space, F)
        d_pre = U_pre.dimension()
        
        if flag_subspaces[d_im] == 0:
            flag_subspaces[d_im] = U_im
            flag_used[d_im] = 1
        
        if flag_subspaces[d_pre] == 0:
            flag_subspaces[d_pre] = U_pre
            flag_used[d_pre] = 1
    
    if test == 1:
        print('(', final_type, ')')
    
    for i in range(0, dim+1):
        if final_type[i] == '?' and final_type[dim - i] != '?':
            i1 = dim - i
            final_type[i] = final_type[i1] - i1 + dim/2
    
    final_type[0] = 0
    
    for i in range(1, dim+1):
        if final_type[i] == '?':
            prev = final_type[i-1]
            if prev != '?' and prev in final_type[i+1:]:
                final_type[i] = prev
                
    for i in range(1, dim+1):
        if final_type[i] == '?':
            final_type[i] = min(final_type[i-1] + 1, dim/2)
    
    if is_final(final_type, dim/2):
        return final_type[1:dim/2 + 1]
    print('error:', final_type[1:dim/2 + 1])
    
def is_final(final_type, dim):
    n = len(final_type)
    if final_type[0] != 0:
        return 0
    
    if final_type[n-1] != dim:
        return 0
    
    for i in range(1, n):
        if final_type[i] != final_type[i - 1] and final_type[i] != final_type[i - 1] + 1:
            return 0
    return 1

from datetime import datetime

now = datetime.now()

current_time = now.strftime("%H:%M:%S")
print("Current Time =", current_time)

p = 17
Rx.<x> = PolynomialRing(GF(p))
C = superelliptic(x^13+x^8+x, 2, p)
print(C.final_type())

now = datetime.now()

current_time = now.strftime("%H:%M:%S")
print("Current Time =", current_time)

Current Time = 18:11:00
[1, 2, 3, 4, 5, 6]
Current Time = 18:11:21

def p_cov(C):
    m = C.exponent
    p = C.characteristic
    f = C.polynomial
    return superelliptic(f(x^p - x), m, p)

p = 3
Rx.<x> = PolynomialRing(GF(p))
C = superelliptic(2*x^9+x^8+x, 2, p)
C1 = p_cov(C)
print(C.final_type(), C1.final_type())

[1, 1, 2, 3] [1, 2, 3, 3, 4, 5, 6, 7, 8, 9, 10, 10, 11]

p = 5
Rx.<x> = PolynomialRing(GF(p))
C = superelliptic(x^9+x^8+x, 2, p)
C1 = p_cov(C)
print(C.final_type(1), C1.final_type(1))

( ['?', 1, 1, 2, 3, 3, 3, 4, 4] )
( ['?', '?', '?', '?', '?', 5, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, '?', 18, 19, 19, '?', 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 22, '?', '?', '?', '?', 22] )
[1, 1, 2, 3] [1, 2, 3, 4, 5, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 18, 18, 19]

C1.genus()

22

p = 11
for a in range(0, p):
    for b in range(0, p):
        for c in range(0, p):
            Rx.<x> = PolynomialRing(GF(p))
            C = superelliptic(x^9+a*x^8+c*x, 2, p)
            if C.is_smooth() and C.p_rank() == 3:
                M = C.cartier_matrix()
                print(C.polynomial, C.p_rank(), (M^4).rank(), 4 - (M-identity_matrix(4)).rank())
                V = C.verschiebung_matrix()
                F = C.frobenius_matrix()
                print(flag(F, V, p))

x^9 + x^8 + x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 2*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 3*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 4*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 5*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 6*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 7*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 8*x 3 0 0
(( [0, 0, 1, 2, 3, 3, 3, 3, 4] ))
[0, 1, 2, 3]
x^9 + x^8 + 9*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 2*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 3*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 4*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 5*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 6*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 7*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 8*x 3 0 0
(( [0, 0, 1, 2, 3, 3, 3, 3, 4] ))
[0, 1, 2, 3]
x^9 + x^8 + 9*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 2*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 3*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 4*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 5*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 6*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 7*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 8*x 3 0 0
(( [0, 0, 1, 2, 3, 3, 3, 3, 4] ))
[0, 1, 2, 3]
x^9 + x^8 + 9*x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + x 3 2 0
(( ['?', '?', 2, 2, 3, 3, 4, '?', 4] ))
[1, 2, 2, 3]
x^9 + x^8 + 2*x 3 2 0

---------------------------------------------------------------------------
KeyboardInterrupt                         Traceback (most recent call last)
<ipython-input-10-442b4b636bbb> in <module>()
      9                 print(C.polynomial, C.p_rank(), (M**Integer(4)).rank(), Integer(4) - (M-identity_matrix(Integer(4))).rank())
     10                 V = C.verschiebung_matrix()
---> 11                 F = C.frobenius_matrix()
     12                 print(flag(F, V, p))

<ipython-input-1-eeab6fc29dc6> in frobenius_matrix(self)
    135         for i in range(Integer(0), len(basis)):
    136             w = basis[i]
--> 137             v = w.frobenius().coordinates()
    138             M[i, :] = v
    139         return M

<ipython-input-1-eeab6fc29dc6> in coordinates(self)
    487                     a = coeff_of_rational_fctn(fct_j, p)
    488                     elt =self - a*superelliptic_cech(C, diffn(G), G)
--> 489                     return elt.coordinates()
    490 
    491         return vector(Integer(2)*g*[Integer(0)])

<ipython-input-1-eeab6fc29dc6> in coordinates(self)
    487                     a = coeff_of_rational_fctn(fct_j, p)
    488                     elt =self - a*superelliptic_cech(C, diffn(G), G)
--> 489                     return elt.coordinates()
    490 
    491         return vector(Integer(2)*g*[Integer(0)])

<ipython-input-1-eeab6fc29dc6> in coordinates(self)
    475                     a = coeff_of_rational_fctn(fct_j, p)
    476                     elt = self - (a/m)*basis[index]
--> 477                     return elt.coordinates() + a/m*vector([GF(p)(i == index) for i in range(Integer(0), Integer(2)*g)])
    478 
    479                 if d<Integer(0):

<ipython-input-1-eeab6fc29dc6> in coordinates(self)
    475                     a = coeff_of_rational_fctn(fct_j, p)
    476                     elt = self - (a/m)*basis[index]
--> 477                     return elt.coordinates() + a/m*vector([GF(p)(i == index) for i in range(Integer(0), Integer(2)*g)])
    478 
    479                 if d<Integer(0):

<ipython-input-1-eeab6fc29dc6> in coordinates(self)
    474                     index = degrees1_inv[(d, j)]
    475                     a = coeff_of_rational_fctn(fct_j, p)
--> 476                     elt = self - (a/m)*basis[index]
    477                     return elt.coordinates() + a/m*vector([GF(p)(i == index) for i in range(Integer(0), Integer(2)*g)])
    478 

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/element.pyx in sage.structure.element.Element.__mul__ (build/cythonized/sage/structure/element.c:12169)()
   1527             if not err:
   1528                 return (<Element>right)._mul_long(value)
-> 1529             return coercion_model.bin_op(left, right, mul)
   1530         except TypeError:
   1531             return NotImplemented

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/coerce.pyx in sage.structure.coerce.CoercionModel.bin_op (build/cythonized/sage/structure/coerce.c:11104)()
   1247             mul_method = getattr(y, '__r%s__'%op_name, None)
   1248             if mul_method is not None:
-> 1249                 res = mul_method(x)
   1250                 if res is not None and res is not NotImplemented:
   1251                     return res

<ipython-input-1-eeab6fc29dc6> in __rmul__(self, constant)
    415         w1 = self.omega0.form
    416         f1 = self.f.function
--> 417         w2 = superelliptic_form(C, constant*w1)
    418         f2 = superelliptic_function(C, constant*f1)
    419         return superelliptic_cech(C, w2, f2)

<ipython-input-1-eeab6fc29dc6> in __init__(self, C, g)
    280         Rxy = PolynomialRing(GF(p), Integer(2), names=('x', 'y',)); (x, y,) = Rxy._first_ngens(2)
    281         Fxy = FractionField(Rxy)
--> 282         g = Fxy(reduction_form(C, g))
    283         self.form = g
    284         self.curve = C

<ipython-input-1-eeab6fc29dc6> in reduction_form(C, g)
    205         else:
    206             G = coff(g, j)
--> 207             g1 += Fxy(y**(j-m)*f*G)
    208     return(g1)
    209 

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/element.pyx in sage.structure.element.Element.__add__ (build/cythonized/sage/structure/element.c:10839)()
   1232         # Left and right are Sage elements => use coercion model
   1233         if BOTH_ARE_ELEMENT(cl):
-> 1234             return coercion_model.bin_op(left, right, add)
   1235 
   1236         cdef long value

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/coerce.pyx in sage.structure.coerce.CoercionModel.bin_op (build/cythonized/sage/structure/coerce.c:10120)()
   1205         # Now coerce to a common parent and do the operation there
   1206         try:
-> 1207             xy = self.canonical_coercion(x, y)
   1208         except TypeError:
   1209             self._record_exception()

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/coerce.pyx in sage.structure.coerce.CoercionModel.canonical_coercion (build/cythonized/sage/structure/coerce.c:11635)()
   1320                 x_elt = x
   1321             if y_map is not None:
-> 1322                 y_elt = (<Map>y_map)._call_(y)
   1323             else:
   1324                 y_elt = y

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/coerce_maps.pyx in sage.structure.coerce_maps.CallableConvertMap._call_ (build/cythonized/sage/structure/coerce_maps.c:7389)()
    430                 y = self._func(C, x)
    431             else:
--> 432                 y = self._func(x)
    433         except Exception:
    434             if print_warnings:

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/rings/fraction_field.py in wrapper(x)
    316 
    317         def wrapper(x):
--> 318             return self._element_class(self, x.numerator(), x.denominator())
    319 
    320         # The case ``S`` being `\QQ` requires special handling since `\QQ` is

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/rings/fraction_field_element.pyx in sage.rings.fraction_field_element.FractionFieldElement_1poly_field.__init__ (build/cythonized/sage/rings/fraction_field_element.c:11744)()
   1115             1/2/x
   1116         """
-> 1117         FractionFieldElement.__init__(self, parent, numerator, denominator,
   1118                 coerce, reduce)
   1119         if not reduce:

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/rings/fraction_field_element.pyx in sage.rings.fraction_field_element.FractionFieldElement.__init__ (build/cythonized/sage/rings/fraction_field_element.c:2966)()
    113         if coerce:
    114             self.__numerator   = parent.ring()(numerator)
--> 115             self.__denominator = parent.ring()(denominator)
    116         else:
    117             self.__numerator   = numerator

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/parent.pyx in sage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9218)()
    898         if mor is not None:
    899             if no_extra_args:
--> 900                 return mor._call_(x)
    901             else:
    902                 return mor._call_with_args(x, args, kwds)

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/coerce_maps.pyx in sage.structure.coerce_maps.NamedConvertMap._call_ (build/cythonized/sage/structure/coerce_maps.c:5949)()
    285             raise TypeError("Cannot coerce {} to {}".format(x, C))
    286         cdef Map m
--> 287         cdef Element e = method(C)
    288         if e is None:
    289             raise RuntimeError("BUG in coercion model: {} method of {} returned None".format(self.method_name, type(x)))

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/rings/polynomial/multi_polynomial.pyx in sage.rings.polynomial.multi_polynomial.MPolynomial._polynomial_ (build/cythonized/sage/rings/polynomial/multi_polynomial.c:6311)()
    205         var = R.variable_name()
    206         if var in self._parent.variable_names():
--> 207             return R(self.polynomial(self._parent(var)))
    208         else:
    209             return R([self])

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/rings/polynomial/multi_polynomial.pyx in sage.rings.polynomial.multi_polynomial.MPolynomial.polynomial (build/cythonized/sage/rings/polynomial/multi_polynomial.c:8657)()
    485         w = {remove_from_tuple(e, ind): val
    486              for e, val in self.dict().iteritems() if not e[ind]}
--> 487         v = [B(w)]  # coefficients that don't involve var
    488         z = var
    489         for i in range(1,d+1):

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/parent.pyx in sage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9218)()
    898         if mor is not None:
    899             if no_extra_args:
--> 900                 return mor._call_(x)
    901             else:
    902                 return mor._call_with_args(x, args, kwds)

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/coerce_maps.pyx in sage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4448)()
    154         cdef Parent C = self._codomain
    155         try:
--> 156             return C._element_constructor(x)
    157         except Exception:
    158             if print_warnings:

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/rings/polynomial/polynomial_ring.py in _element_constructor_(self, x, check, is_gen, construct, **kwds)
    466         elif isinstance(x, sage.rings.power_series_ring_element.PowerSeries):
    467             x = x.truncate()
--> 468         return C(self, x, check, is_gen, construct=construct, **kwds)
    469 
    470     @classmethod

src/cysignals/signals.pyx in cysignals.signals.python_check_interrupt()

KeyboardInterrupt: 

C = superelliptic(x^9+x^8+x, 2, 11)
print(C.basis_holomorphic_differentials)

[(1/y) dx, (x/y) dx, (x^2/y) dx, (x^3/y) dx]

p = 5
Rx.<x> = PolynomialRing(GF(p))
C = superelliptic(x^3 - x, 2, p)
if (C.is_smooth()):
    print(C.p_rank())
    print(C.final_type())

1
[1]

C.frobenius_matrix()

[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0]
[0 0 0 4 2 3 3 2]
[4 0 4 4 2 0 0 4]
[0 0 4 3 0 0 1 2]
[3 0 0 0 0 1 0 0]

flag(F, V, p)

test [0, -1, -1, -1, 4, -1, -1, -1, 4]
poprawiony ft [0, 1, 2, 3, 4, 4, 4, 4, 4]

[1, 2, 3, 4]

f = x^9+3*x^3+x^2+x+1
r = f.degree()
i = 2
j = 1
m = 2
print(Rx(m-j)*Rx(x)*Rx(f.derivative()))
print(Rx(m)*Rx(i)*f)

4*x^9 + 4*x^3 + 2*x^2 + x
4*x^9 + 2*x^3 + 4*x^2 + 4*x + 4

print(r*(m-j))
print(m*i)
print(f.derivative())

9
4
4*x^8 + 4*x^2 + 2*x + 1

x*f.derivative() - 2*2*f

2*x^3 + 3*x^2 + 2*x + 1

M = matrix(QQ, [[1,2], [3,6]])
U = M.kernel()
V = VectorSpace(QQ,2)
M.transpose().image()

Vector space of degree 2 and dimension 1 over Rational Field
Basis matrix:
[1 3]

l = U.basis()
l = l +[(1, 1/3)]

###fragment kodu do obliczania residuuow w niesk - zaniechany
#def more_v(f, prec):
#    C = f.curve
#    f = f.vw
#    g = C.polynomial8
#    p = C.characteristic
#    m = C.exponent
#    r = C.polynomial.degree()
#    delta, a, b = xgcd(m, r)
#    a = -a
#    M = m/delta
#    R = r/delta
#    
#    Fpbar = GF(p).algebraic_closure()
#    Ruv.<u, v> = PolynomialRing(Fpbar, 2)
#    if prec == 0:
#        return 0
#    zeta = Fpbar.zeta(m)
#    a = f(v = zeta, w = 0)
#    f1 = f - a
#    if w.divides(f1):
#        return more_v(f1/w, prec-1)
    

p = 7
Rx.<x> = PolynomialRing(GF(p))
C = superelliptic(x^3 + x + 3, 5, p)
baza = C.basis_de_rham
print(C.genus())
#E = EllipticCurve(GF(p), [1, 2])
print(E.trace_of_frobenius())
#C.basis_holomorphic_differentials(  )

4
2

C.degree_de_rham0

{0: (0, 2),
 1: (0, 3),
 2: (0, 4),
 3: (1, 4),
 4: (-1, 1),
 5: (0, 1),
 6: (1, 2),
 7: (1, 3)}

C.basis_de_rham

{0: ((1/y^2) dx, 0, (1/y^2) dx),
 1: ((1/y^3) dx, 0, (1/y^3) dx),
 2: ((1/y^4) dx, 0, (1/y^4) dx),
 3: ((x/y^4) dx, 0, (x/y^4) dx),
 4: (0 dx, 5/x*y^4, ((x + 1)/(x^2*y)) dx),
 5: ((2/y) dx, 5/x^2*y^4, ((-x + 2)/(x^3*y)) dx),
 6: (((-3*x)/y^2) dx, 5/x*y^3, ((2*x + 1)/(x^2*y^2)) dx),
 7: ((x/y^3) dx, 5/x*y^2, ((3*x + 1)/(x^2*y^3)) dx)}

A = C.frobenius_matrix()
B = C.verschiebung_matrix()

A.solve_right((0, 0, 0, 0, 4, 2, 5, 3))

(1, 0, 0, 4, 3, 0, 2, 0)

A*vector((1, 0, 0, 4, 3, 0, 2, 0))

(0, 0, 0, 0, 4, 2, 5, 3)

vector((1, 0, 0, 4, 5, 0, 0, 0)) - vector((0, 0, 0, 0, 2, 6, 5, 5))

(1, 0, 0, 4, 3, -6, -5, -5)

omega = diffn(superelliptic_function(C, y^2))

omega.jth_component(0)

3*x^2 + 1

R.<x, y> = PolynomialRing(GF(p), 2)
g1 = x^3*y^7 + x^2*y^9
g2 = x^2*y + y^6
R1.<x> = PolynomialRing(GF(p))
R2 = FractionField(R1)
R3.<y> = PolynomialRing(R2)

xgcd(R3(g1), R3(g2))[1]*R3(g1) + xgcd(R3(g1), R3(g2))[2]*R3(g2)

y

H = HyperellipticCurve(x^5 - x + 1)

H

Hyperelliptic Curve over Finite Field of size 5 defined by y^2 = x^5 + 4*x + 1

f = x^3 + x + 2

f.derivative(x)

-2*x^2 + 1

p = 5
R1.<x> = PolynomialRing(GF(p))
R2 = FractionField(R1)
R3.<y> = PolynomialRing(R2)
g = y^2/x + y/(x+1)    
g = 1/y+x/y^2

R3.<z> = PolynomialRing(R2)
g(y = 1/z)

x*z^2 + z

f

x^3 + x + 4

f.coefficient()

---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
<ipython-input-62-e054c182ec1a> in <module>()
----> 1 f.coefficient()

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/element.pyx in sage.structure.element.Element.__getattr__ (build/cythonized/sage/structure/element.c:4614)()
    485             AttributeError: 'LeftZeroSemigroup_with_category.element_class' object has no attribute 'blah_blah'
    486         """
--> 487         return self.getattr_from_category(name)
    488 
    489     cdef getattr_from_category(self, name):

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/element.pyx in sage.structure.element.Element.getattr_from_category (build/cythonized/sage/structure/element.c:4723)()
    498         else:
    499             cls = P._abstract_element_class
--> 500         return getattr_from_other_class(self, cls, name)
    501 
    502     def __dir__(self):

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/cpython/getattr.pyx in sage.cpython.getattr.getattr_from_other_class (build/cythonized/sage/cpython/getattr.c:2614)()
    392         dummy_error_message.cls = type(self)
    393         dummy_error_message.name = name
--> 394         raise AttributeError(dummy_error_message)
    395     attribute = <object>attr
    396     # Check for a descriptor (__get__ in Python)

AttributeError: 'sage.rings.polynomial.polynomial_zmod_flint.Polynomial_zmod_flint' object has no attribute 'coefficient'

x^3+x+1

x^3 + x + 1

parent(x)

Symbolic Ring

R.<x> = PolynomialRing(GF(5))

R = (x^3+x).parent()

R.<x, y> = PolynomialRing(GF(5))
RR = FractionField(R)
A = RR(1/(x*y))

A.derivative(x)

(-1)/(x^2*y)

dict1 = {}
dict1[3] = 5
dict1[6] = 121

degrees1_inv = {b:a for a, b in dict1.items()}

degrees1_inv

{5: 3, 121: 6}

C

Superelliptic curve with the equation y^7 = x^3 + x + 2 over finite field with 5 elements.

basis = C.basis_de_rham()

basis.items()

dict_items([(0, ((x/y) dx, 2/x*y, ((x^3*y^5 - x^3 + x - 1)/(x^2*y^6)) dx)), (1, (((-1)/y) dx, 2/x^2*y, ((-x^3*y^5 + x^3 - 2*x - 2)/(x^3*y^6)) dx)), (2, (((-2*x)/y^2) dx, 2/x*y^2, ((-2*x^3*y^3 + x^3 - 1)/(x^2*y^5)) dx)), (3, ((1/y^2) dx, 2/x^2*y^2, ((x^3*y^3 - 2*x^3 + 2*x - 2)/(x^3*y^5)) dx)), (4, ((1/y^3) dx, 0, (1/y^3) dx)), (5, (0 dx, 2/x*y^3, ((-2*x^3 - x - 1)/(x^2*y^4)) dx)), (6, ((1/y^4) dx, 0, (1/y^4) dx)), (7, ((2*x/y^4) dx, 2/x*y^4, ((2*x^3 - 2*x*y - y)/(x^2*y^4)) dx)), (8, ((1/y^5) dx, 0, (1/y^5) dx)), (9, ((x/y^5) dx, 0, (x/y^5) dx)), (10, ((1/y^6) dx, 0, (1/y^6) dx)), (11, ((x/y^6) dx, 0, (x/y^6) dx))])

m = 9
r = 6
delta, a, b = xgcd(m, r)
a = -a
xgcd(9, 6)

(3, 1, -1)

b*r -a*m, delta

(3, 3)

Fpbar = GF(5).algebraic_closure()
z = Fpbar.zeta(7)

GF(5^6)(z)

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-23-42625790d9c3> in <module>()
----> 1 GF(Integer(5)**Integer(6))(z)

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/parent.pyx in sage.structure.parent.Parent.__call__ (build/cythonized/sage/structure/parent.c:9218)()
    898         if mor is not None:
    899             if no_extra_args:
--> 900                 return mor._call_(x)
    901             else:
    902                 return mor._call_with_args(x, args, kwds)

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/coerce_maps.pyx in sage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4556)()
    159                 print(type(C), C)
    160                 print(type(C._element_constructor), C._element_constructor)
--> 161             raise
    162 
    163     cpdef Element _call_with_args(self, x, args=(), kwds={}):

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/structure/coerce_maps.pyx in sage.structure.coerce_maps.DefaultConvertMap_unique._call_ (build/cythonized/sage/structure/coerce_maps.c:4448)()
    154         cdef Parent C = self._codomain
    155         try:
--> 156             return C._element_constructor(x)
    157         except Exception:
    158             if print_warnings:

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/rings/finite_rings/finite_field_givaro.py in _element_constructor_(self, e)
    368             2*a4^3 + 2*a4^2 + 1
    369         """
--> 370         return self._cache.element_from_data(e)
    371 
    372     def gen(self, n=0):

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/rings/finite_rings/element_givaro.pyx in sage.rings.finite_rings.element_givaro.Cache_givaro.element_from_data (build/cythonized/sage/rings/finite_rings/element_givaro.cpp:7458)()
    312         return make_FiniteField_givaroElement(self,res)
    313 
--> 314     cpdef FiniteField_givaroElement element_from_data(self, e):
    315         """
    316         Coerces several data types to ``self``.

/opt/sagemath-9.1/local/lib/python3.7/site-packages/sage/rings/finite_rings/element_givaro.pyx in sage.rings.finite_rings.element_givaro.Cache_givaro.element_from_data (build/cythonized/sage/rings/finite_rings/element_givaro.cpp:7080)()
    451 
    452         else:
--> 453             raise TypeError("unable to coerce %r" % type(e))
    454 
    455         cdef GEN t

TypeError: unable to coerce <class 'sage.rings.algebraic_closure_finite_field.AlgebraicClosureFiniteField_pseudo_conway_with_category.element_class'>

Integers(7)(5).multiplicative_order()

6

Rx.<x> = PolynomialRing(QQ)
f = sum((i+1)*x^i for i in range(0, 10))

cut(f, 5)

10*x^3 + 9*x^2 + 8*x + 7

f

10*x^9 + 9*x^8 + 8*x^7 + 7*x^6 + 6*x^5 + 5*x^4 + 4*x^3 + 3*x^2 + 2*x + 1

E = EllipticCurve(GF(2), [0,0,1,0,1])
print(E)
print(E.frobenius_polynomial())

Elliptic Curve defined by y^2 + y = x^3 + 1 over Finite Field of size 2
x^2 + 2

E

Elliptic Curve defined by y^2 + y = x^3 + x over Finite Field of size 2

x^2 + 2*x + 2