# -*- coding: utf-8 -*-
r"""
Projective plane conics over a rational function field

The class :class:`ProjectiveConic_rational_function_field` represents a
projective plane conic over a rational function field `F(t)`, where `F`
is any field. Instances can be created using :func:`Conic`.

AUTHORS:

- Lennart Ackermans (2016-02-07): initial version

EXAMPLES:

Create a conic::

    sage: K = FractionField(PolynomialRing(QQ, 't'))
    sage: P.<X, Y, Z> = K[]
    sage: Conic(X^2 + Y^2 - Z^2)
    Projective Conic Curve over Fraction Field of Univariate
    Polynomial Ring in t over Rational Field defined by
    X^2 + Y^2 - Z^2

Points can be found using :meth:`has_rational_point`::

    sage: K.<t> = FractionField(QQ['t'])
    sage: C = Conic([1,-t,t])
    sage: C.has_rational_point(point = True)
    (True, (0 : 1 : 1))
"""

# ****************************************************************************
#       Copyright (C) 2016 Lennart Ackermans
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
#                  https://www.gnu.org/licenses/
# ****************************************************************************

from sage.rings.all import PolynomialRing
from sage.matrix.constructor import diagonal_matrix, matrix, block_matrix
from sage.schemes.plane_conics.con_field import ProjectiveConic_field
from sage.arith.all import lcm, gcd
from sage.modules.free_module_element import vector
from sage.rings.fraction_field import is_FractionField


class ProjectiveConic_rational_function_field(ProjectiveConic_field):
    r"""
    Create a projective plane conic curve over a rational function field
    `F(t)`, where `F` is any field.

    The algorithms used in this class come mostly from [HC2006]_.

    EXAMPLES::

        sage: K = FractionField(PolynomialRing(QQ, 't'))
        sage: P.<X, Y, Z> = K[]
        sage: Conic(X^2 + Y^2 - Z^2)
        Projective Conic Curve over Fraction Field of Univariate
        Polynomial Ring in t over Rational Field defined by
        X^2 + Y^2 - Z^2

    TESTS::

        sage: K = FractionField(PolynomialRing(QQ, 't'))
        sage: Conic([K(1), 1, -1])._test_pickling()

    REFERENCES:

    - [HC2006]_
    - [Ack2016]_
    """
    def __init__(self, A, f):
        r"""
        See ``Conic`` for full documentation.

        EXAMPLES::

            sage: c = Conic([1, 1, 1]); c
            Projective Conic Curve over Rational Field defined by
            x^2 + y^2 + z^2
        """
        ProjectiveConic_field.__init__(self, A, f)

    def has_rational_point(self, point=False, algorithm='default',
                           read_cache=True):
        r"""
        Returns True if and only if the conic ``self``
        has a point over its base field `F(t)`, which is a field of rational
        functions.

        If ``point`` is True, then returns a second output, which is
        a rational point if one exists.

        Points are cached whenever they are found. Cached information
        is used if and only if ``read_cache`` is True.

        The default algorithm does not (yet) work for all base fields `F`.
        In particular, sage is required to have:

        * an algorithm for finding the square root of elements in finite
          extensions of `F`;

        * a factorization and gcd algorithm for `F[t]`;

        * an algorithm for solving conics over `F`.

        ALGORITHM:

        The parameter ``algorithm`` specifies the algorithm
        to be used:

        * ``'default'`` -- use a native Sage implementation, based on the
          algorithm Conic in [HC2006]_.

        * ``'magma'`` (requires Magma to be installed) --
          delegates the task to the Magma computer algebra
          system.

        EXAMPLES:

        We can find points for function fields over (extensions of) `\QQ`
        and finite fields::

            sage: K.<t> = FractionField(PolynomialRing(QQ, 't'))
            sage: C = Conic(K, [t^2-2, 2*t^3, -2*t^3-13*t^2-2*t+18])
            sage: C.has_rational_point(point=True)
            (True, (-3 : (t + 1)/t : 1))
            sage: R.<t> = FiniteField(23)[]
            sage: C = Conic([2, t^2+1, t^2+5])
            sage: C.has_rational_point()
            True
            sage: C.has_rational_point(point=True)
            (True, (5*t : 8 : 1))
            sage: F.<i> = QuadraticField(-1)
            sage: R.<t> = F[]
            sage: C = Conic([1,i*t,-t^2+4])
            sage: C.has_rational_point(point = True)
            (True, (-t - 2*i : -2*i : 1))

        It works on non-diagonal conics as well::

            sage: K.<t> = QQ[]
            sage: C = Conic([4, -4, 8, 1, -4, t + 4])
            sage: C.has_rational_point(point=True)
            (True, (1/2 : 1 : 0))

        If no point exists output still depends on the argument ``point``::

            sage: K.<t> = QQ[]
            sage: C = Conic(K, [t^2, (t-1), -2*(t-1)])
            sage: C.has_rational_point()
            False
            sage: C.has_rational_point(point=True)
            (False, None)

        Due to limitations in Sage of algorithms we depend on, it is not
        yet possible to find points on conics over multivariate function fields
        (see the requirements above)::

            sage: F.<t1> = FractionField(QQ['t1'])
            sage: K.<t2> = FractionField(F['t2'])
            sage: a = K(1)
            sage: b = 2*t2^2+2*t1*t2-t1^2
            sage: c = -3*t2^4-4*t1*t2^3+8*t1^2*t2^2+16*t1^3-t2-48*t1^4
            sage: C = Conic([a,b,c])
            sage: C.has_rational_point()
            Traceback (most recent call last):
            ...
            NotImplementedError: is_square() not implemented for elements of
            Univariate Quotient Polynomial Ring in tbar over Fraction Field
            of Univariate Polynomial Ring in t1 over Rational Field with
            modulus tbar^2 + t1*tbar - 1/2*t1^2

        In some cases, the algorithm requires us to be
        able to solve conics over `F`. In particular, the following does not
        work::

            sage: P.<u> = QQ[]
            sage: E = P.fraction_field()
            sage: Q.<Y> = E[]
            sage: F.<v> = E.extension(Y^2 - u^3 - 1)
            sage: R.<t> = F[]
            sage: K = R.fraction_field()
            sage: C = Conic(K, [u, v, 1])
            sage: C.has_rational_point()
            Traceback (most recent call last):
            ...
            NotImplementedError: has_rational_point not implemented for conics
            over base field Univariate Quotient Polynomial Ring in v over
            Fraction Field of Univariate Polynomial Ring in u over Rational
            Field with modulus v^2 - u^3 - 1

        ``has_rational_point`` fails for some conics over function fields
        over finite fields, due to :trac:`20003`::

            sage: K.<t> = PolynomialRing(GF(7))
            sage: C = Conic([5*t^2+4, t^2+3*t+3, 6*t^2+3*t+2, 5*t^2+5, 4*t+3, 4*t^2+t+5])
            sage: C.has_rational_point()
            Traceback (most recent call last):
            ...
            TypeError: self (=Scheme morphism:
              From: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 defined by (5*t^2 + 4)*x^2 + ((6*t^3 + 3*t^2 + 5*t + 5)/(t + 3))*y^2 + ((6*t^6 + 3*t^5 + t^3 + 6*t^2 + 6*t + 2)/(t^4 + t^3 + 4*t^2 + 3*t + 1))*z^2
              To:   Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 defined by (5*t^2 + 4)*x^2 + (t^2 + 3*t + 3)*x*y + (5*t^2 + 5)*y^2 + (6*t^2 + 3*t + 2)*x*z + (4*t + 3)*y*z + (4*t^2 + t + 5)*z^2
              Defn: Defined on coordinates by sending (x : y : z) to
                    (x + ((2*t + 5)/(t + 3))*y + ((3*t^4 + 2*t^3 + 5*t^2 + 5*t + 3)/(t^4 + t^3 + 4*t^2 + 3*t + 1))*z : y + ((6*t^3 + 6*t^2 + 3*t + 6)/(t^3 + 4*t^2 + 2*t + 2))*z : z)) domain must equal right (=Scheme morphism:
              From: Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 defined by (5*t^3 + 6*t^2 + 3*t + 3)*x^2 + (t + 4)*y^2 + (6*t^7 + 2*t^5 + t^4 + 2*t^3 + 3*t^2 + 6*t + 6)*z^2
              To:   Projective Conic Curve over Fraction Field of Univariate Polynomial Ring in t over Finite Field of size 7 defined by (5/(t^3 + 4*t^2 + 2*t + 2))*x^2 + (1/(t^3 + 3*t^2 + 5*t + 1))*y^2 + ((6*t^6 + 3*t^5 + t^3 + 6*t^2 + 6*t + 2)/(t^9 + 5*t^8 + t^7 + 6*t^6 + 3*t^5 + 4*t^3 + t^2 + 5*t + 3))*z^2
              Defn: Defined on coordinates by sending (x : y : z) to
                    ((t^3 + 4*t^2 + 2*t + 2)*x : (t^2 + 5)*y : (t^5 + 4*t^4 + t^2 + 3*t + 3)*z)) codomain


        TESTS::

            sage: K.<t> = FractionField(PolynomialRing(QQ, 't'))
            sage: a = (2*t^2 - 3/2*t + 1)/(37/3*t^2 + t - 1/4)
            sage: b = (1/2*t^2 + 1/3)/(-73*t^2 - 2*t + 11/4)
            sage: c = (6934/3*t^6 + 8798/3*t^5 - 947/18*t^4 + 3949/9*t^3 + 20983/18*t^2 + 28/3*t - 131/3)/(-2701/3*t^4 - 293/3*t^3 + 301/6*t^2 + 13/4*t - 11/16)
            sage: C = Conic([a,b,c])
            sage: C.has_rational_point(point=True)
            (True, (4*t + 4 : 2*t + 2 : 1))

        A long time test::

            sage: K.<t> = FractionField(PolynomialRing(QQ, 't'))
            sage: a = (-1/3*t^6 - 14*t^5 - 1/4*t^4 + 7/2*t^2 - 1/2*t - 1)/(24/5*t^6 - t^5 - 1/4*t^4 + t^3 - 3*t^2 + 8/5*t + 5)
            sage: b = (-3*t^3 + 8*t + 1/2)/(-1/3*t^3 + 3/2*t^2 + 1/12*t + 1/2)
            sage: c = (1232009/225*t^25 - 1015925057/8100*t^24 + 1035477411553/1458000*t^23 + 7901338091/30375*t^22 - 1421379260447/729000*t^21 + 266121260843/972000*t^20 + 80808723191/486000*t^19 - 516656082523/972000*t^18 + 21521589529/40500*t^17 + 4654758997/21600*t^16 - 20064038625227/9720000*t^15 - 173054270347/324000*t^14 + 536200870559/540000*t^13 - 12710739349/50625*t^12 - 197968226971/135000*t^11 - 134122025657/810000*t^10 + 22685316301/120000*t^9 - 2230847689/21600*t^8 - 70624099679/270000*t^7 - 4298763061/270000*t^6 - 41239/216000*t^5 - 13523/36000*t^4 + 493/36000*t^3 + 83/2400*t^2 + 1/300*t + 1/200)/(-27378/125*t^17 + 504387/500*t^16 - 97911/2000*t^15 + 1023531/4000*t^14 + 1874841/8000*t^13 + 865381/12000*t^12 + 15287/375*t^11 + 6039821/6000*t^10 + 599437/1500*t^9 + 18659/250*t^8 + 1218059/6000*t^7 + 2025127/3000*t^6 + 1222759/6000*t^5 + 38573/200*t^4 + 8323/125*t^3 + 15453/125*t^2 + 17031/500*t + 441/10)
            sage: C = Conic([a,b,c])
            sage: C.has_rational_point(point = True) # long time (4 seconds)
            (True,
             ((-2/117*t^8 + 304/1053*t^7 + 40/117*t^6 - 1/27*t^5 - 110/351*t^4 - 2/195*t^3 + 11/351*t^2 + 1/117)/(t^4 + 2/39*t^3 + 4/117*t^2 + 2/39*t + 14/39) : -5/3*t^4 + 19*t^3 : 1))
        """
        from .constructor import Conic

        if read_cache:
            if self._rational_point is not None:
                return (True, self._rational_point) if point else True

        if algorithm != 'default':
            return ProjectiveConic_field.has_rational_point(self, point,
                algorithm, read_cache)

        # Default algorithm
        if self.base_ring().characteristic() == 2:
            raise NotImplementedError("has_rational_point not implemented \
for function field of characteristic 2.")
        new_conic, transformation, inverse = self.diagonalization()
        coeff = new_conic.coefficients()
        if coeff[0] == 0:
            return (True, transformation([1, 0, 0])) if point else True
        elif coeff[3] == 0:
            return (True, transformation([0, 1, 0])) if point else True
        elif coeff[5] == 0:
            return (True, transformation([0, 0, 1])) if point else True

        # We save the coefficients of the reduced form in coeff
        # A zero of the reduced conic can be multiplied by multipliers
        # to get a zero of the old conic
        (coeff, multipliers) = new_conic._reduce_conic()
        new_conic = Conic(coeff)
        transformation = transformation \
            * new_conic.hom(diagonal_matrix(multipliers))
        if coeff[0].degree() % 2 == coeff[1].degree() % 2 and \
                coeff[1].degree() % 2 == coeff[2].degree() % 2:
            case = 0
        else:
            case = 1

        t, = self.base_ring().base().gens()  # t in F[t]
        supp = []
        roots = [[], [], []]
        remove = None
        # loop through the coefficients and find a root of f_i (as in
        # [HC2006]) modulo each element in the coefficients' support
        for i in (0, 1, 2):
            supp.append(list(coeff[i].factor()))
            for p in supp[i]:
                if p[1] != 1:
                    raise ValueError("Expected factor of exponent 1.")
                # Convert to monic factor
                x = p[0] / list(p[0])[-1]
                N = p[0].base_ring().extension(x, 'tbar')
                R = PolynomialRing(N, 'u')
                u, = R.gens()
                # If p[0] has degree 1, sage might forget the "defining
                # polynomial" of N, so we define our own modulo operation
                if p[0].degree() == 1:
                    mod = t.parent().hom([-x[0]])
                else:
                    mod = N
                if i == 0:
                    x = -mod(coeff[2]) / mod(coeff[1])
                elif i == 1:
                    x = -mod(coeff[0]) / mod(coeff[2])
                else:
                    x = -mod(coeff[1]) / mod(coeff[0])
                if x.is_square():
                    root = N(x.sqrt())
                else:
                    return (False, None) if point else False
                # if case == 0 and p[0] has degree 1, we switch to case
                # 1 and remove this factor out of the support. In [HC2006]
                # this is done later, in FindPoint.
                if case == 0 and p[0].degree() == 1:
                    case = 1
                    # remove later so the loop iterator stays in place.
                    remove = (i, p)
                else:
                    roots[i].append(root)
        if remove:
            supp[remove[0]].remove(remove[1])
        supp = [[p[0] for p in supp[i]] for i in (0, 1, 2)]

        if case == 0:
            # Find a solution of (5) in [HC2006]
            leading_conic = Conic(self.base_ring().base_ring(),
                        [coeff[0].leading_coefficient(),
                        coeff[1].leading_coefficient(),
                        coeff[2].leading_coefficient()])
            has_point = leading_conic.has_rational_point(True)
            if has_point[0]:
                if point:
                    pt = new_conic.find_point(supp, roots, case,
                        has_point[1])
                else:
                    pt = True
                return (True, transformation(pt)) if point else True
            else:
                return (False, None) if point else False
        # case == 1:
        if point:
            pt = new_conic.find_point(supp, roots, case)
        else:
            pt = True
        return (True, transformation(pt)) if point else True

    def _reduce_conic(self):
        r"""
        Return the reduced form of the conic, i.e. a conic with base field
        `K=F(t)` and coefficients `a,b,c` such that `a,b,c \in F[t]`,
        `\gcd(a,b)=\gcd(b,c)=\gcd(c,a)=1` and `abc` is square-free.

        Assumes `self` is in diagonal form.

        OUTPUT:

        A tuple (coefficients, multipliers), the coefficients of the conic
        in reduced form and multipliers `\lambda, \mu, \nu \in F(t)^*` such
        that `(x,y,z) \in F(t)` is a solution of the reduced conic if and only
        if `(\lambda x, \mu y, \nu z)` is a solution of `self`.

        ALGORITHM:

        The algorithm used is the algorithm ReduceConic in [HC2006]_.

        EXAMPLES::

            sage: K.<t> = FractionField(PolynomialRing(QQ, 't'))
            sage: C = Conic(K, [t^2-2, 2*t^3, -2*t^3-13*t^2-2*t+18])
            sage: C._reduce_conic()
            ([t^2 - 2, 2*t, -2*t^3 - 13*t^2 - 2*t + 18], [t, 1, t])
        """

        # start with removing fractions
        coeff = [self.coefficients()[0], self.coefficients()[3],
                self.coefficients()[5]]
        coeff = lcm(lcm(coeff[0].denominator(), coeff[1].denominator()),
                coeff[2].denominator()) * vector(coeff)
        # go to base ring of fraction field
        coeff = [self.base().base()(x) for x in coeff]
        coeff = vector(coeff) / gcd(coeff)
        # remove common divisors
        labda = mu = nu = 1
        g1 = g2 = g3 = 0
        ca, cb, cc = coeff
        while g1 != 1 or g2 != 1 or g3 != 1:
            g1 = gcd(ca, cb)
            ca = ca / g1
            cb = cb / g1
            cc = cc * g1
            nu = g1 * nu
            g2 = gcd(ca, cc)
            ca = ca / g2
            cc = cc / g2
            cb = cb * g2
            mu = g2 * mu
            g3 = gcd(cb, cc)
            cb = cb / g3
            cc = cc / g3
            ca = ca * g3
            labda = g3 * labda
        coeff = [ca, cb, cc]
        multipliers = [labda, mu, nu]

        # remove squares
        for i, x in enumerate(coeff):
            if is_FractionField(x.parent()):
                # go to base ring of fraction field
                x = self.base().base()(x)

            try:
                decom = x.squarefree_decomposition()
            except (NotImplementedError, AttributeError):
                decom = x.factor()
            x = decom.unit()
            x2 = 1
            for factor in decom:
                if factor[1] > 1:
                    if factor[1] % 2 == 0:
                        x2 *= factor[0] ** (factor[1] // 2)
                    else:
                        x *= factor[0]
                        x2 *= factor[0] ** ((factor[1] - 1) // 2)
                else:
                    x *= factor[0]
            for j, y in enumerate(multipliers):
                if j != i:
                    multipliers[j] = y * x2
            coeff[i] = self.base_ring().base().coerce(x)

        return (coeff, multipliers)

    def find_point(self, supports, roots, case, solution=0):
        r"""
        Given a solubility certificate like in [HC2006]_, find a point on
        ``self``. Assumes ``self`` is in reduced form (see [HC2006]_ for a
        definition).

        If you don't have a solubility certificate and just want to find a
        point, use the function :meth:`has_rational_point` instead.

        INPUT:

        - ``self`` -- conic in reduced form.
        - ``supports`` -- 3-tuple where ``supports[i]`` is a list of all monic
          irreducible `p \in F[t]` that divide the `i`'th of the 3 coefficients.
        - ``roots`` -- 3-tuple containing lists of roots of all elements of
          ``supports[i]``, in the same order.
        - ``case`` -- 1 or 0, as in [HC2006]_.
        - ``solution`` -- (default: 0) a solution of (5) in [HC2006]_, if
          case = 0, 0 otherwise.

        OUTPUT:

        A point `(x,y,z) \in F(t)` of ``self``. Output is undefined when the
        input solubility certificate is incorrect.

        ALGORITHM:

        The algorithm used is the algorithm FindPoint in [HC2006]_, with
        a simplification from [Ack2016]_.

        EXAMPLES::

            sage: K.<t> = FractionField(QQ['t'])
            sage: C = Conic(K, [t^2-2, 2*t^3, -2*t^3-13*t^2-2*t+18])
            sage: C.has_rational_point(point=True) # indirect test
            (True, (-3 : (t + 1)/t : 1))

        Different solubility certificates give different points::

            sage: K.<t> = PolynomialRing(QQ, 't')
            sage: C = Conic(K, [t^2-2, 2*t, -2*t^3-13*t^2-2*t+18])
            sage: supp = [[t^2 - 2], [t], [t^3 + 13/2*t^2 + t - 9]]
            sage: tbar1 = QQ.extension(supp[0][0], 'tbar').gens()[0]
            sage: tbar2 = QQ.extension(supp[1][0], 'tbar').gens()[0]
            sage: tbar3 = QQ.extension(supp[2][0], 'tbar').gens()[0]
            sage: roots = [[tbar1 + 1], [1/3*tbar2^0], [2/3*tbar3^2 + 11/3*tbar3 - 3]]
            sage: C.find_point(supp, roots, 1)
            (3 : t + 1 : 1)
            sage: roots = [[-tbar1 - 1], [-1/3*tbar2^0], [-2/3*tbar3^2 - 11/3*tbar3 + 3]]
            sage: C.find_point(supp, roots, 1)
            (3 : -t - 1 : 1)
        """
        Ft = self.base().base()
        F = Ft.base()
        t, = Ft.gens()
        coefficients = [Ft(self.coefficients()[0]), Ft(self.coefficients()[3]),
            Ft(self.coefficients()[5])]
        deg = [coefficients[0].degree(), coefficients[1].degree(),
               coefficients[2].degree()]
        # definitions as in [HC2006] and [Ack2016]
        A = ((deg[1] + deg[2]) / 2).ceil() - case
        B = ((deg[2] + deg[0]) / 2).ceil() - case
        C = ((deg[0] + deg[1]) / 2).ceil() - case

        # For all roots as calculated by has_rational_point(), we create
        # a system of linear equations. As in [Ack2016], we do this
        # by calculating the matrices for all phi_p, with basis consisting
        # of monomials of x, y and z in the space V of potential solutions:
        # t^0, ..., t^A, t^0, ..., t^B and t^0, ..., t^C.
        phi = []
        for (i, p) in enumerate(supports[0]):
            # lift to F[t] and map to R, with R as defined above
            if roots[0][i].parent().is_finite():
                root = roots[0][i].polynomial()
            else:
                root = roots[0][i].lift()
            alpha = root.parent().hom([t])(root)
            d = p.degree()
            # Calculate y - alpha*z mod p for all basis vectors
            phi_p = [[] for i in range(A + B + C + 4)]
            phi_p[0:A + 1] = [vector(F, d)] * (A + 1)
            phi_p[A + 1] = vector(F, d, {0: F(1)})
            lastpoly = F(1)
            for n in range(B):
                lastpoly = (lastpoly * t) % p
                phi_p[A + 2 + n] = vector(F, d, lastpoly.dict())
            lastpoly = -alpha % p
            phi_p[A + B + 2] = vector(F, d, lastpoly.dict())
            for n in range(C):
                lastpoly = (lastpoly * t) % p
                phi_p[A + B + 3 + n] = vector(F, d, lastpoly.dict())
            phi_p[A + B + C + 3] = vector(F, d)
            phi.append(matrix(phi_p).transpose())
        for (i, p) in enumerate(supports[1]):
            if roots[1][i].parent().is_finite():
                root = roots[1][i].polynomial()
            else:
                root = roots[1][i].lift()
            alpha = root.parent().hom([t])(root)
            d = p.degree()
            # Calculate z - alpha*x mod p for all basis vectors
            phi_p = [[] for i in range(A + B + C + 4)]
            phi_p[A + 1:A + B + 2] = [vector(F, d)] * (B + 1)
            phi_p[A + B + 2] = vector(F, d, {0: F(1)})
            lastpoly = F(1)
            for n in range(C):
                lastpoly = (lastpoly * t) % p
                phi_p[A + B + 3 + n] = vector(F, d, lastpoly.dict())
            lastpoly = -alpha % p
            phi_p[0] = vector(F, d, lastpoly.dict())
            for n in range(A):
                lastpoly = (lastpoly * t) % p
                phi_p[1 + n] = vector(F, d, lastpoly.dict())
            phi_p[A + B + C + 3] = vector(F, d)
            phi.append(matrix(phi_p).transpose())
        for (i, p) in enumerate(supports[2]):
            if roots[2][i].parent().is_finite():
                root = roots[2][i].polynomial()
            else:
                root = roots[2][i].lift()
            alpha = root.parent().hom([t])(root)
            d = p.degree()
            # Calculate x - alpha*y mod p for all basis vectors
            phi_p = [[] for i in range(A + B + C + 4)]
            phi_p[A + B + 2:A + B + C + 3] = [vector(F, d)] * (C + 1)
            phi_p[0] = vector(F, d, {0: F(1)})
            lastpoly = F(1)
            for n in range(A):
                lastpoly = (lastpoly * t) % p
                phi_p[1 + n] = vector(F, d, lastpoly.dict())
            lastpoly = -alpha % p
            phi_p[A + 1] = vector(F, d, lastpoly.dict())
            for n in range(B):
                lastpoly = (lastpoly * t) % p
                phi_p[A + 2 + n] = vector(F, d, lastpoly.dict())
            phi_p[A + B + C + 3] = vector(F, d)
            phi.append(matrix(phi_p).transpose())
        if case == 0:
            # We need three more equations
            lx = Ft(solution[0]).leading_coefficient()
            ly = Ft(solution[1]).leading_coefficient()
            lz = Ft(solution[2]).leading_coefficient()
            ABC = A + B + C
            phi.append(matrix([vector(F, ABC + 4, {A: 1, ABC + 3: -lx}),
                vector(F, ABC + 4, {A + B + 1: 1, ABC + 3: -ly}),
                vector(F, ABC + 4, {ABC + 2: 1, ABC + 3: -lz})]))
        # Create the final matrix which we will solve
        M = block_matrix(phi, ncols=1, subdivide=False)
        solution_space = M.right_kernel()
        for v in solution_space.basis():
            if v[:A + B + C + 3] != 0:
                # we do not want to return a trivial solution
                X = Ft(list(v[:A + 1]))
                Y = Ft(list(v[A + 1:A + B + 2]))
                Z = Ft(list(v[A + B + 2:A + B + C + 3]))
                return self.point([X, Y, Z])

        raise RuntimeError("No solution has been found: possibly incorrect "
                           "solubility certificate.")
