From 5ea79c7ae0d44686f1df05c4a016652afbe58968 Mon Sep 17 00:00:00 2001 From: Sebastiano Tronto Date: Sun, 14 Jun 2026 09:58:21 +0200 Subject: Initial commit --- misc_code/2_division.sage | 126 + misc_code/2_division_counterex.sage | 59 + misc_code/2_division_onecurve.sage | 101 + misc_code/2_division_withparisplit.sage | 122 + misc_code/Q49-49a1-49a2.gp | 14 + misc_code/adelic_failure_example | 3 + misc_code/adelic_failure_example.sage | 10 + misc_code/ugly_computations/GL2_ZN.py | 24 + misc_code/ugly_computations/cm/cm-rank+-all.sage | 3361 ++++++++++++++++++++ misc_code/ugly_computations/cm/test-cm-rank.sage | 16 + misc_code/ugly_computations/order_GLn.pdf | 2137 +++++++++++++ .../ugly_computations/quadratic_class_number.sage | 12 + 12 files changed, 5985 insertions(+) create mode 100755 misc_code/2_division.sage create mode 100755 misc_code/2_division_counterex.sage create mode 100755 misc_code/2_division_onecurve.sage create mode 100755 misc_code/2_division_withparisplit.sage create mode 100755 misc_code/Q49-49a1-49a2.gp create mode 100755 misc_code/adelic_failure_example create mode 100755 misc_code/adelic_failure_example.sage create mode 100755 misc_code/ugly_computations/GL2_ZN.py create mode 100755 misc_code/ugly_computations/cm/cm-rank+-all.sage create mode 100755 misc_code/ugly_computations/cm/test-cm-rank.sage create mode 100755 misc_code/ugly_computations/order_GLn.pdf create mode 100755 misc_code/ugly_computations/quadratic_class_number.sage (limited to 'misc_code') diff --git a/misc_code/2_division.sage b/misc_code/2_division.sage new file mode 100755 index 0000000..68cea80 --- /dev/null +++ b/misc_code/2_division.sage @@ -0,0 +1,126 @@ + +R1. = PolynomialRing(QQ) +R2. = PolynomialRing(QQ) + +def extended_field( f, A, B, deg_mult ): + # f: a polynomial whose roots are the x-coordinates of some points of an + # elliptic curve E: y^2 = x^3 + Ax + B. + # return value: a field containing the x and y coordinates of those points + # deg_mult: a positive integer known to be a multiple of the degree of the + # extended field. + # + # This function uses the properties of resultants (I can provide a pdf + # explaining how it works). + # + # It is useful to compute, e.g., the fields obtained by adjoining the + # coordinates of the n-division points of a point (using the n-uplication + # formulas to get the required polynomials). + # + # When used to compute the 2-division fields, it gives the same output as + # E.division_field(2). + + + g = y^2 - x^3 - A*x - B + res = f.resultant(g,x) + res = res.subs(y=x) + + #K. = f.splitting_field() + #print aux + print "+++ Computing splitting field of the following: +++" + + pol = R1(res*f) + print pol + + K. = NumberField( R( pari(pol).nfsplitting(deg_mult) ) ) + + return K + +L = [] + +for A in range(1,9): + for B in range(1,9): + + print "Current list of examples:", len(L), "elements. List:" + print L + + E = EllipticCurve([0,0,0,A,B]) + print "*************************" + print E + print "of rank ", E.rank(), "with (some) points of infinite order:", E.gens() + print "CM:", E.has_cm() + + rep = E.galois_representation() + print "mod 2 rep is surjective:", rep.is_surjective(2) + + if E.rank() == 0: + print "Stopping because rank 0" + print "" + continue + if len(E.gens()) == 0: + print "Stopping because no points of infinite order found" + print "" + continue + if E.has_cm(): + print "Stopping because CM" + print "" + print "" + continue + if not rep.is_surjective(2): + print "Stopping because mod 2 rep is not surjective" + print "" + continue + + K_2. = E.division_field(2) + print "2-division field:", K_2 + P = E(0) + flag = False + for P in E.gens(): + if len(P.division_points(2)) == 0: + flag = True + break + if not flag: + print "Stopping because the points found are 2-divisible" + print "" + continue + print "Taking the 2-division of P =", P + + # The following polynomial is derived from the duplication formula + # (Silverman, p.54) using the x-coordinate of the 2-division point as + # an indeterminate.x + f_P = R1(x^4 - 4*P[0]*x^3 - 2*A*x^2 - 4*(2*B + A*P[0])*x + A^2 - 4*B*P[0]) + #print f_P.roots() + + M = extended_field( f_P, A, B, 24 ) # The 2-division field of P + print "2-division field of P:", M + + if M.degree() != 24: + print "Stopping because 2-division of P is too small" + print "" + continue + + div_pol_4 = E.division_polynomial(4) + + #if div_pol_4.splitting_field('zz').degree() < 48: + # print "Stopping because splitting field of div_pol_4 < 48" + # exit() + + K_4 = extended_field( div_pol_4, A, B, 96 ) + print "4-division field:", K_4 + + if K_4.degree() != 96: + print "Stopping because mod 4 representation not surjective" + print "" + continue + + if len(f_P.roots(ring=K_4)) == 0: + print "Stopping because M is not contained in K_4" + print "" + continue + + print "----------------" + print "|Example Found!|" + print "----------------" + L.append((A,B)) + + print "*************************" + print "" diff --git a/misc_code/2_division_counterex.sage b/misc_code/2_division_counterex.sage new file mode 100755 index 0000000..f1193fb --- /dev/null +++ b/misc_code/2_division_counterex.sage @@ -0,0 +1,59 @@ + +R1. = PolynomialRing(QQ) +R2. = PolynomialRing(QQ) + +def extended_field( f, A, B, deg_mult ): + # f: a polynomial whose roots are the x-coordinates of some points of an + # elliptic curve E: y^2 = x^3 + Ax + B. + # return value: a field containing the x and y coordinates of those points + # deg_mult: a positive integer known to be a multiple of the degree of the + # extended field. + # + # This function uses the properties of resultants (I can provide a pdf + # explaining how it works). + # + # It is useful to compute, e.g., the fields obtained by adjoining the + # coordinates of the n-division points of a point (using the n-uplication + # formulas to get the required polynomials). + # + # When used to compute the 2-division fields, it gives the same output as + # E.division_field(2). + + + g = y^2 - x^3 - A*x - B + res = f.resultant(g,x) + res = res.subs(y=x) + + #print "Splitting field of division pol:" + #print R1(f).splitting_field('r') + K. = (R1(res*f)).splitting_field(degree_multiple=deg_mult) + + return K + + +A = -3 +B = 17/4 + +E = EllipticCurve([0,0,0,A,B]) +print E +print "of rank ", E.rank(), "with (some) points of infinite order:", E.gens() +print "CM:", E.has_cm() + +rep = E.galois_representation() +print "mod 2 rep is surjective:", rep.is_surjective(2) + +K_2. = E.division_field(2) +print "2-division field:", K_2 +P = [4,15/2] + +# The following polynomial is derived from the duplication formula +# (Silverman, p.54) using the x-coordinate of the 2-division point as +# an indeterminate.x +f_P = R1(x^4 - 4*P[0]*x^3 - 2*A*x^2 - 4*(2*B + A*P[0])*x + A^2 - 4*B*P[0]) + +M = extended_field( f_P, A, B, 24 ) # The 2-division field of P +print "2-division field of P:", M + +if M.degree() != 24: + print "Stopping because 2-division of P is too small" + exit() diff --git a/misc_code/2_division_onecurve.sage b/misc_code/2_division_onecurve.sage new file mode 100755 index 0000000..9cb7043 --- /dev/null +++ b/misc_code/2_division_onecurve.sage @@ -0,0 +1,101 @@ + +R1. = PolynomialRing(QQ) +R2. = PolynomialRing(QQ) + +def extended_field( f, A, B, deg_mult ): + # f: a polynomial whose roots are the x-coordinates of some points of an + # elliptic curve E: y^2 = x^3 + Ax + B. + # return value: a field containing the x and y coordinates of those points + # deg_mult: a positive integer known to be a multiple of the degree of the + # extended field. + # + # This function uses the properties of resultants (I can provide a pdf + # explaining how it works). + # + # It is useful to compute, e.g., the fields obtained by adjoining the + # coordinates of the n-division points of a point (using the n-uplication + # formulas to get the required polynomials). + # + # When used to compute the 2-division fields, it gives the same output as + # E.division_field(2). + + + g = y^2 - x^3 - A*x - B + res = f.resultant(g,x) + res = res.subs(y=x) + + #print "Splitting field of division pol:" + #print R1(f).splitting_field('r') + K. = (R1(res*f)).splitting_field(degree_multiple=deg_mult) + + return K + + +A = -3 +B = 17/4 + +E = EllipticCurve([0,0,0,A,B]) +print "*************************" +print E +print "of rank ", E.rank(), "with (some) points of infinite order:", E.gens() +print "CM:", E.has_cm() + +rep = E.galois_representation() +print "mod 2 rep is surjective:", rep.is_surjective(2) + +if E.rank() == 0: + print "Stopping because rank 0" + exit() +if len(E.gens()) == 0: + print "Stopping because no points of infinite order found" + exit() +if E.has_cm(): + print "Stopping because CM" + print "" + exit() +if not rep.is_surjective(2): + print "Stopping because mod 2 rep is not surjective" + exit() + +K_2. = E.division_field(2) +print "2-division field:", K_2 +P = E(0) +flag = False +for P in E.gens(): + if len(P.division_points(2)) == 0: + flag = True + break +if not flag: + print "Stopping because the points found are 2-divisible" + exit() +print "Taking the 2-division of P =", P + +# The following polynomial is derived from the duplication formula +# (Silverman, p.54) using the x-coordinate of the 2-division point as +# an indeterminate.x +f_P = R1(x^4 - 4*P[0]*x^3 - 2*A*x^2 - 4*(2*B + A*P[0])*x + A^2 - 4*B*P[0]) + +M = extended_field( f_P, A, B, 24 ) # The 2-division field of P +print "2-division field of P:", M + +if M.degree() != 24: + print "Stopping because 2-division of P is too small" + exit() + +K_4 = extended_field( E.division_polynomial(4), A, B, 96 ) +print "4-division field:", K_4 + +if K_4.degree() != 96: + print "Stopping because mod 4 representation not surjective" + exit() + +if len(f_P.roots(ring=K_4)) == 0: + print "Stopping because E is not contained in K_4" + exit() + +print "----------------" +print "|Example Found!|" +print "----------------" + +print "*************************" +print "" diff --git a/misc_code/2_division_withparisplit.sage b/misc_code/2_division_withparisplit.sage new file mode 100755 index 0000000..349ea95 --- /dev/null +++ b/misc_code/2_division_withparisplit.sage @@ -0,0 +1,122 @@ + +R1. = PolynomialRing(QQ) +R2. = PolynomialRing(QQ) + +def extended_field( f, A, B, deg_mult ): + # f: a polynomial whose roots are the x-coordinates of some points of an + # elliptic curve E: y^2 = x^3 + Ax + B. + # return value: a field containing the x and y coordinates of those points + # deg_mult: a positive integer known to be a multiple of the degree of the + # extended field. + # + # This function uses the properties of resultants (I can provide a pdf + # explaining how it works). + # + # It is useful to compute, e.g., the fields obtained by adjoining the + # coordinates of the n-division points of a point (using the n-uplication + # formulas to get the required polynomials). + # + # When used to compute the 2-division fields, it gives the same output as + # E.division_field(2). + + + g = y^2 - x^3 - A*x - B + res = f.resultant(g,x) + res = res.subs(y=x) + + #K. = f.splitting_field() + #print aux + pol = pari(R1(res*f)).nfsplitting(deg_mult) + K. = NumberField(g) + + return K + +L = [] + +for A in range(1,9): + for B in range(1,9): + + print "Current list of examples:", len(L), "elements. List:" + print L + + E = EllipticCurve([0,0,0,A,B]) + print "*************************" + print E + print "of rank ", E.rank(), "with (some) points of infinite order:", E.gens() + print "CM:", E.has_cm() + + rep = E.galois_representation() + print "mod 2 rep is surjective:", rep.is_surjective(2) + + if E.rank() == 0: + print "Stopping because rank 0" + print "" + continue + if len(E.gens()) == 0: + print "Stopping because no points of infinite order found" + print "" + continue + if E.has_cm(): + print "Stopping because CM" + print "" + print "" + continue + if not rep.is_surjective(2): + print "Stopping because mod 2 rep is not surjective" + print "" + continue + + K_2. = E.division_field(2) + print "2-division field:", K_2 + P = E(0) + flag = False + for P in E.gens(): + if len(P.division_points(2)) == 0: + flag = True + break + if not flag: + print "Stopping because the points found are 2-divisible" + print "" + continue + print "Taking the 2-division of P =", P + + # The following polynomial is derived from the duplication formula + # (Silverman, p.54) using the x-coordinate of the 2-division point as + # an indeterminate.x + f_P = R1(x^4 - 4*P[0]*x^3 - 2*A*x^2 - 4*(2*B + A*P[0])*x + A^2 - 4*B*P[0]) + #print f_P.roots() + + M = extended_field( f_P, A, B, 24 ) # The 2-division field of P + print "2-division field of P:", M + + if M.degree() != 24: + print "Stopping because 2-division of P is too small" + print "" + continue + + div_pol_4 = E.division_polynomial(4) + + #if div_pol_4.splitting_field('zz').degree() < 48: + # print "Stopping because splitting field of div_pol_4 < 48" + # exit() + + K_4 = extended_field( div_pol_4, A, B, 96 ) + print "4-division field:", K_4 + + if K_4.degree() != 96: + print "Stopping because mod 4 representation not surjective" + print "" + continue + + if len(f_P.roots(ring=K_4)) == 0: + print "Stopping because M is not contained in K_4" + print "" + continue + + print "----------------" + print "|Example Found!|" + print "----------------" + L.append((A,B)) + + print "*************************" + print "" diff --git a/misc_code/Q49-49a1-49a2.gp b/misc_code/Q49-49a1-49a2.gp new file mode 100755 index 0000000..a8ae14d --- /dev/null +++ b/misc_code/Q49-49a1-49a2.gp @@ -0,0 +1,14 @@ +elltorsfield_deg( E, n, deg ) = { + my( f, eqn, res ); + f = elldivpol( E, n ); + eqn = y^2 + E[1]*x*y + E[3]*y - x^3 - E[2]*x^2 - E[4]*x - E[5]; + res = substpol( polresultant( f, eqn ), y, x ); + printf( "Computing %d-torsion field...\n", n ); + return( nfinit( nfsplitting( f*res, deg ) ) ); +} + +print("Computing 49-torsion field for 49a1, expecting maximal degree 42*49\n"); +elltorsfield_deg( ellinit( "49a1" ), 49, 42*49 ); + +print("Computing 49-torsion field for 49a1, expecting maximal degree 42*49\n"); +elltorsfield_deg( ellinit( "49a1" ), 49, 42*49 ); diff --git a/misc_code/adelic_failure_example b/misc_code/adelic_failure_example new file mode 100755 index 0000000..d84180e --- /dev/null +++ b/misc_code/adelic_failure_example @@ -0,0 +1,3 @@ +E = EllipticCurve([0, 1, 0, -44, -84]) +print "Rational torsion points:", E.torsion_points() +print "Generator of the free part:", E.gens() diff --git a/misc_code/adelic_failure_example.sage b/misc_code/adelic_failure_example.sage new file mode 100755 index 0000000..1fdca8c --- /dev/null +++ b/misc_code/adelic_failure_example.sage @@ -0,0 +1,10 @@ +E = EllipticCurve([0, 1, 0, -44, -84]) +print "Rational torsion points:", E.torsion_points() +print "Generators of the free part:", E.gens() +P = E.gens()[0] + +K = CyclotomicField(3) +EK = E.base_extend(K) +PK = EK.point(P) + +print "2-division points over Q(zeta_3):", PK.division_points(2) diff --git a/misc_code/ugly_computations/GL2_ZN.py b/misc_code/ugly_computations/GL2_ZN.py new file mode 100755 index 0000000..c706703 --- /dev/null +++ b/misc_code/ugly_computations/GL2_ZN.py @@ -0,0 +1,24 @@ +# Prints a list of matrices, the elements of GL_2(Z/nZ) + +def gcd( m, n ): + if n == 0: + return m + return gcd( n, m%n ) + +def print_mat( a, b, c, d ): + print "" + print "|", a, " ", b, "|" + print "|", c, " ", d, "|" + +n = input("Choose n: ") + +count = 0 +for a in range(n): + for b in range(n): + for c in range(n): + for d in range(n): + if gcd( n, abs(a*d-b*c)) == 1: + print_mat(a,b,c,d) + count += 1 + +print "Found", count, "elements" diff --git a/misc_code/ugly_computations/cm/cm-rank+-all.sage b/misc_code/ugly_computations/cm/cm-rank+-all.sage new file mode 100755 index 0000000..f1943af --- /dev/null +++ b/misc_code/ugly_computations/cm/cm-rank+-all.sage @@ -0,0 +1,3361 @@ + +# Elliptic curves downloaded from the LMFDB downloaded on 23 October 2019. +# Below is a list called data. Each entry has the form: +# [a1,a2,a3,a4,a6] (Weierstrass Coefficients) + + +data = [\ +[0, -1, 1, -887, -10143],\ +[0, -1, 1, -7, 10],\ +[0, 0, 1, 0, -34],\ +[0, 0, 1, 0, 1],\ +[0, 0, 1, 0, -1],\ +[0, 0, 1, 0, 20],\ +[0, 1, 0, -13, -21],\ +[0, 1, 0, -3, 1],\ +[0, 0, 0, -2, 0],\ +[0, 0, 0, 8, 0],\ +[0, 0, 0, -12, 0],\ +[0, 0, 0, 3, 0],\ +[0, 0, 1, -13718, -619025],\ +[0, 0, 1, -38, 90],\ +[0, 0, 0, 0, -4],\ +[0, 0, 0, 0, 108],\ +[1, -1, 1, -16400, -804212],\ +[1, -1, 1, -965, -13940],\ +[1, -1, 1, -335, 2440],\ +[1, -1, 1, -20, 46],\ +[0, 0, 1, 0, -331],\ +[0, 0, 1, 0, 12],\ +[0, 0, 0, -396, -3024],\ +[0, 0, 0, -396, 3024],\ +[0, 0, 0, -36, 0],\ +[0, 0, 0, 9, 0],\ +[0, 0, 0, -540, -4752],\ +[0, 0, 0, -60, 176],\ +[0, 0, 0, 0, -216],\ +[0, 0, 0, 0, 8],\ +[0, 0, 1, -6750, -213469],\ +[0, 0, 1, -750, 7906],\ +[0, 0, 1, 0, -844],\ +[0, 0, 1, 0, 31],\ +[0, 0, 0, -29155, -1915998],\ +[0, 0, 0, -1715, -33614],\ +[0, 0, 0, -595, 5586],\ +[0, 0, 0, -35, 98],\ +[0, 0, 0, -275, -1750],\ +[0, 0, 0, -275, 1750],\ +[0, 0, 0, -25, 0],\ +[0, 0, 0, 100, 0],\ +[0, 0, 0, -5, 0],\ +[0, 0, 0, 20, 0],\ +[0, 0, 0, 0, -2700],\ +[0, 0, 0, 0, 100],\ +[0, 0, 0, 0, -972],\ +[0, 0, 0, 0, 36],\ +[0, 0, 0, 0, -243],\ +[0, 0, 0, 0, 9],\ +[0, 0, 1, 0, -2],\ +[0, 0, 1, 0, 47],\ +[0, 0, 1, 0, -16207],\ +[0, 0, 1, 0, 600],\ +[0, 0, 1, 0, -192787],\ +[0, 0, 1, 0, 7140],\ +[0, 0, 1, 0, -88],\ +[0, 0, 1, 0, 3],\ +[0, 0, 0, -539, -4802],\ +[0, 0, 0, -539, 4802],\ +[0, 0, 0, -49, 0],\ +[0, 0, 0, 196, 0],\ +[0, 0, 0, -500, 0],\ +[0, 0, 0, 125, 0],\ +[0, 0, 0, -20, 0],\ +[0, 0, 0, 5, 0],\ +[0, 0, 0, -1080, -13662],\ +[0, 0, 0, -120, 506],\ +[0, 0, 0, 0, -54],\ +[0, 0, 0, 0, 2],\ +[0, 0, 0, -1080, 13662],\ +[0, 0, 0, -120, -506],\ +[0, 0, 0, 0, -2],\ +[0, 0, 0, 0, 54],\ +[0, 0, 0, 0, -28],\ +[0, 0, 0, 0, 756],\ +[0, 0, 0, -6615, 203742],\ +[0, 0, 0, -735, -7546],\ +[0, 0, 0, 0, -343],\ +[0, 0, 0, 0, 9261],\ +[0, 0, 1, -1590140, -771794326],\ +[0, 0, 1, -860, 9707],\ +[0, 0, 0, -72, 0],\ +[0, 0, 0, 18, 0],\ 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