CoCalc -- Field norm.sagews

Project: test

Views: ¹²⁴

M = ModularForms(Gamma0(3),2)
M.set_precision(14)
M.basis()

[
1 + 12*q + 36*q^2 + 12*q^3 + 84*q^4 + 72*q^5 + 36*q^6 + 96*q^7 + 180*q^8 + 12*q^9 + 216*q^10 + 144*q^11 + 84*q^12 + 168*q^13 + O(q^14)
]

EllipticCurve([0,0,1,0,-61])
phi = EllipticCurve([0,0,1,0,-61]).modular_parametrization()
print(phi)
phi.power_series(prec = 100)
print(parent(phi))

Elliptic Curve defined by y^2 + y = x^3 - 61 over Rational Field
Modular parameterization from the upper half plane to Elliptic Curve defined by y^2 + y = x^3 - 61 over Rational Field
(q^-2 + q + 8*q^4 - 19*q^7 + 43*q^10 - 70*q^13 + 160*q^16 - 324*q^19 + 739*q^22 - 1473*q^25 + 3000*q^28 - 5855*q^31 + 11635*q^34 - 22901*q^37 + 45184*q^40 - 88149*q^43 + 171524*q^46 - 331954*q^49 + 641584*q^52 - 1236088*q^55 + 2376836*q^58 - 4557893*q^61 + 8723040*q^64 - 16660735*q^67 + 31769685*q^70 - 60481943*q^73 + 114976336*q^76 - 218264380*q^79 + 413813619*q^82 - 783621456*q^85 + 1482270144*q^88 - 2800885598*q^91 + 5287367236*q^94 - 9972031158*q^97 + O(q^98), -q^-3 - 2 + 18*q^3 - 23*q^6 + 81*q^9 - 171*q^12 + 511*q^15 - 1215*q^18 + 2925*q^21 - 6303*q^24 + 13851*q^27 - 29727*q^30 + 64308*q^33 - 135999*q^36 + 285669*q^39 - 591601*q^42 + 1219374*q^45 - 2494755*q^48 + 5080932*q^51 - 10285461*q^54 + 20724210*q^57 - 41558899*q^60 + 83021112*q^63 - 165233601*q^66 + 327779385*q^69 - 648177471*q^72 + 1278087300*q^75 - 2513373628*q^78 + 4930361865*q^81 - 9649221597*q^84 + 18843716794*q^87 - 36724618674*q^90 + 71436441195*q^93 - 138708240983*q^96 + O(q^97))
<type 'instance'>

Q.<x> = QQ[]
P = x^2-5
K.<w> = P.splitting_field(); K
OK.basis()

OK = K.ring_of_integers()
I = OK.ideal(2)
I
I.factor()

L.<a,b> = NumberField([x^2-5,x^2-13])
O = L.ring_of_integers()
O.basis()
J = O.ideal(2)
J.factor()
J = O.ideal(3); J.factor()

Number Field in w with defining polynomial x^2 - 5
[1/2*w + 1/2, w]
Fractional ideal (2)
Fractional ideal (2)
[3/2*a - b + 1/2, 3*a - 2*b, (-1/4*b + 11/4)*a - 7/4*b + 9/4, 11/2*a - 7/2*b]
(Fractional ideal (1/2*a + 1/2*b + 1)) * (Fractional ideal ((-1/4*b + 3/4)*a - 1/4*b + 7/4))
(Fractional ideal (1/2*b - 1/2)) * (Fractional ideal (1/2*b + 1/2))

a = -7; b = -12
-4*a^3-27*b^2
for a in [-40..-1] :
   for b in [1..40] :
      D=-4*a^3-27*b^2
      d = sqrt(D)
      if d.is_integer() :
         [a,b,D,d]

-2516
[-39, 19, 227529, 477]
[-39, 26, 219024, 468]
[-37, 37, 165649, 407]
[-31, 30, 94864, 308]
[-27, 27, 59049, 243]
[-21, 7, 35721, 189]
[-21, 17, 29241, 171]
[-21, 20, 26244, 162]
[-21, 28, 15876, 126]
[-21, 35, 3969, 63]
[-21, 37, 81, 9]
[-19, 19, 17689, 133]
[-19, 30, 3136, 56]
[-13, 12, 4900, 70]
[-13, 13, 4225, 65]
[-12, 8, 5184, 72]
[-12, 16, 0, 0]
[-9, 9, 729, 27]
[-7, 6, 400, 20]
[-7, 7, 49, 7]
[-3, 1, 81, 9]
[-3, 2, 0, 0]

Q.<x> = QQ[]
#P = x^4+x^3+x^2+x+1
P = x^3-39*x-19
P.factor()



K.<w> = P.splitting_field(); K

type(K)
OK = K.ring_of_integers()
base = OK.basis(); base
M.<u,v,t> = ZZ[]

OK.class_group()

    
G = K.galois_group();

t = [[0,0,0],[0,0,0],[0,0,0]];
for m in [0..len(base)-1] :
    for n in [0..len(G)-1] :
      t[m][n] = (G[1]^n)(base[m])
    
t


I = OK.ideal(2,w)
type(I)
type(K.pari_nf())

R.<a,b,c> = K[]
T = [a,b,c]
no = 1;
for n in [0..len(G)-1] :
    tmp = 0;
    for m in [0..len(base)-1] :
       tmp = tmp+t[m][n]*T[m]
    no = no*tmp

no
no.factor()

x^3 - 39*x - 19
Number Field in w with defining polynomial x^3 - 39*x - 19
<class 'sage.rings.number_field.number_field.NumberField_absolute_with_category'>
[2/53*w^2 + 23/53*w + 1/53, w, w^2]
Class group of order 1 of Number Field in w with defining polynomial x^3 - 39*x - 19
[[2/53*w^2 + 23/53*w + 1/53, -7/53*w^2 - 1/53*w + 235/53, 5/53*w^2 - 22/53*w - 77/53], [w, -13/53*w^2 - 17/53*w + 338/53, 13/53*w^2 - 36/53*w - 338/53], [w^2, -36/53*w^2 + 169/53*w + 2314/53, -17/53*w^2 - 169/53*w + 1820/53]]
<class 'sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal'>
<type 'sage.libs.pari.gen.gen'>
a^3 + 9*a^2*b + 24*a*b^2 + 19*b^3 - 21*a^2*c - 57*a*b*c - 252*a*c^2 - 741*b*c^2 + 361*c^3
(a + (3/53*w^2 + 8/53*w + 81/53)*b + (-32/53*w^2 - 103/53*w + 461/53)*c) * (a + (-4/53*w^2 + 7/53*w + 263/53)*b + (47/53*w^2 - 69/53*w - 1593/53)*c) * (a + (1/53*w^2 - 15/53*w + 133/53)*b + (-15/53*w^2 + 172/53*w + 19/53)*c)