notebook.community

Edit and run



In [1]:

    
1+1









    Out[1]:





2



In [2]:

    
20/3









    Out[2]:





6



In [3]:

    
20.0/3









    Out[3]:





6.666666666666667



In [6]:

    
type (20.0 / 3)









    Out[6]:





float

Let's load SageMath's hooks



In [7]:

    
%load_ext sage



In [8]:

    
1+1









    Out[8]:





2



In [9]:

    
type(20.0/3)









    Out[9]:





<type 'sage.rings.real_mpfr.RealNumber'>

Algebraic Structures



In [10]:

    
ZZ, QQ, RR, CC









    Out[10]:





(Integer Ring,
 Rational Field,
 Real Field with 53 bits of precision,
 Complex Field with 53 bits of precision)



In [11]:

    
Zmod(30)









    Out[11]:





Ring of integers modulo 30



In [12]:

    
GF(31)









    Out[12]:





Finite Field of size 31

Most algebraic structures are organised as parent/elements



In [13]:

    
GF(17).list()









    Out[13]:





[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]



In [14]:

    
a = 2



In [15]:

    
a.parent()









    Out[15]:





Integer Ring



In [16]:

    
ZZ is a.parent()









    Out[16]:





True



In [17]:

    
a.is_unit()









    Out[17]:





False

Conversions



In [18]:

    
b = QQ(a)



In [19]:

    
b == a









    Out[19]:





True



In [20]:

    
b.parent()









    Out[20]:





Rational Field



In [21]:

    
b.is_unit()









    Out[21]:





True

Working with notebooks



In [22]:

    
a, b = 2, 3



In [23]:

    
c = a + b



In [24]:

    
c









    Out[24]:





5

Exploring objects

Completion



In [ ]:

    
ZZ.<tab>

Documentation



In [ ]:

    
ZZ.CartesianProduct?



In [ ]:

    
ZZ.cardinality??

Polynomial rings



In [33]:

    
a = x^2 - 1



In [34]:

    
b = (x-1)*(x+1)
b









    Out[34]:





(x + 1)*(x - 1)



In [35]:

    
a == b









    Out[35]:





x^2 - 1 == (x + 1)*(x - 1)



In [36]:

    
bool(a == b)









    Out[36]:





True

x is just a variable name



In [37]:

    
x = 2



In [38]:

    
x^2 - 1









    Out[38]:





3

Forget about x!

We don't need symbolic calculus, we need polynomials We can reset symbolic variables with x = SR.var('x')



In [39]:

    
QQ['x']









    Out[39]:





Univariate Polynomial Ring in x over Rational Field



In [40]:

    
R.<x> = QQ[]
R









    Out[40]:





Univariate Polynomial Ring in x over Rational Field



In [41]:

    
a = x^2 -1
a









    Out[41]:





x^2 - 1



In [42]:

    
a.parent()









    Out[42]:





Univariate Polynomial Ring in x over Rational Field



In [43]:

    
b = (x-1)*(x+1)
b









    Out[43]:





x^2 - 1



In [44]:

    
a == b









    Out[44]:





True

Finite fields



In [46]:

    
p = next_prime(2^20)
R.<x> = GF(p)[]
R









    Out[46]:





Univariate Polynomial Ring in x over Finite Field of size 1048583



In [48]:

    
P = R.irreducible_element(20)
P









    Out[48]:





x^20 + x^19 + 1048569*x^18 + 1048575*x^17 + 99*x^16 + 44*x^15 + 1048224*x^14 + 1048469*x^13 + 907*x^12 + 268*x^11 + 1047424*x^10 + 1048551*x^9 + 1552*x^8 + 299*x^7 + 1047653*x^6 + 1047736*x^5 + 3036*x^4 + 38*x^3 + 172*x^2 + 1048279*x + 991



In [49]:

    
K.<z> = GF(p^20, modulus=P)
K









    Out[49]:





Finite Field in z of size 1048583^20



In [50]:

    
z^20









    Out[50]:





1048582*z^19 + 14*z^18 + 8*z^17 + 1048484*z^16 + 1048539*z^15 + 359*z^14 + 114*z^13 + 1047676*z^12 + 1048315*z^11 + 1159*z^10 + 32*z^9 + 1047031*z^8 + 1048284*z^7 + 930*z^6 + 847*z^5 + 1045547*z^4 + 1048545*z^3 + 1048411*z^2 + 304*z + 1047592

If we don't care about the modulus



In [51]:

    
L.<z> = GF(p^2)
L









    Out[51]:





Finite Field in z of size 1048583^2



In [52]:

    
L.modulus()









    Out[52]:





x^2 + x + 1



In [53]:

    
z.multiplicative_order()









    Out[53]:





3



In [54]:

    
z^3









    Out[54]:





1



In [56]:

    
e = L.primitive_element()
e.multiplicative_order()









    Out[56]:





1099526307888



In [57]:

    
e









    Out[57]:





z + 5



In [58]:

    
e.minpoly()









    Out[58]:





x^2 + 1048574*x + 21

Elliptic Curves



In [59]:

    
E = EllipticCurve([1,2])
E









    Out[59]:





Elliptic Curve defined by y^2 = x^3 + x + 2 over Rational Field



In [60]:

    
E.j_invariant()









    Out[60]:





432/7



In [61]:

    
E.torsion_order()









    Out[61]:





4



In [62]:

    
E.torsion_points()









    Out[62]:





[(-1 : 0 : 1), (0 : 1 : 0), (1 : -2 : 1), (1 : 2 : 1)]



In [64]:

    
G = E.torsion_subgroup()
G









    Out[64]:





Torsion Subgroup isomorphic to Z/4 associated to the Elliptic Curve defined by y^2 = x^3 + x + 2 over Rational Field



In [65]:

    
G.gens()









    Out[65]:





((1 : 2 : 1),)

Elliptic curves over finite fields



In [67]:

    
E = EllipticCurve(GF(p), [0, 1, 0, 2, 3])
E









    Out[67]:





Elliptic Curve defined by y^2 = x^3 + x^2 + 2*x + 3 over Finite Field of size 1048583



In [68]:

    
E.cardinality().factor()









    Out[68]:





3^3 * 7 * 23 * 241



In [70]:

    
P = E.random_point()
Q = E.lift_x(123)
P, Q









    Out[70]:





((866576 : 1022542 : 1), (123 : 70573 : 1))



In [71]:

    
P+Q









    Out[71]:





(986279 : 792082 : 1)



In [72]:

    
Q.order().factor()









    Out[72]:





3^3 * 23 * 241



In [75]:

    
R = 9*23*241*Q
R









    Out[75]:





(1048581 : 818907 : 1)

Isogenies



In [77]:

    
phi = E.isogeny([R])
phi









    Out[77]:





Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + x^2 + 2*x + 3 over Finite Field of size 1048583 to Elliptic Curve defined by y^2 = x^3 + x^2 + 1048485*x + 343 over Finite Field of size 1048583



In [78]:

    
S = phi(P)
S









    Out[78]:





(871094 : 464333 : 1)



In [79]:

    
S.parent()









    Out[79]:





Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x^2 + 1048485*x + 343 over Finite Field of size 1048583



In [80]:

    
phi(R)









    Out[80]:





(0 : 1 : 0)



In [81]:

    
phi.rational_maps()









    Out[81]:





((x^3 + 4*x^2 + 24*x + 20)/(x^2 + 4*x + 4),
 (x^3*y + 6*x^2*y - 8*x*y + 8*y)/(x^3 + 6*x^2 + 12*x + 8))



In [82]:

    
phi.kernel_polynomial()









    Out[82]:





x + 2



In [83]:

    
phi.kernel_polynomial()(R[0])









    Out[83]:





0

Pairings



In [89]:

    
L.<z> = GF(p^2)
EE = E.change_ring(L)
EE









    Out[89]:





Elliptic Curve defined by y^2 = x^3 + x^2 + 2*x + 3 over Finite Field in z of size 1048583^2



In [90]:

    
EE.cardinality().factor()









    Out[90]:





3^4 * 7 * 19 * 23 * 241 * 18413



In [92]:

    
T = EE.lift_x(912851*z + 87136)
T.order()









    Out[92]:





3



In [93]:

    
w = T.weil_pairing(EE(R), 3)
w









    Out[93]:





1048582*z + 1048582



In [94]:

    
w.multiplicative_order()









    Out[94]:





3



In [95]:

    
T.weil_pairing(2*T, 3)









    Out[95]:





1



In [ ]: