{{{id=1|
R.<x,y,z>=GF(4,'a')[]
C=Curve(x^2+y^2+z^2+x*y+y*z+x*z)
///
}}}

{{{id=2|
C.rational_points()
///
[(0 : a : 1), (0 : a + 1 : 1), (1 : 1 : 1), (a : 0 : 1), (a : 1 : 0), (a : a + 1 : 1), (a + 1 : 0 : 1), (a + 1 : 1 : 0), (a + 1 : a : 1)]
}}}

{{{id=3|
C.irreducible_components()
///
[
Closed subscheme of Projective Space of dimension 2 over Finite Field in a of size 2^2 defined by:
  x + (a)*y + (a + 1)*z,
Closed subscheme of Projective Space of dimension 2 over Finite Field in a of size 2^2 defined by:
  x + (a + 1)*y + (a)*z
]
}}}

{{{id=15|
R.<x,y,z>=QQ[]
f=y^2*z-x^3-x^2*z-x*z^2-z^3
print f
print f.subs(x=x-z/3)
g=f.subs(x=(x-3*z),z=9*z,y=y/3)
print g
///
-x^3 - x^2*z + y^2*z - x*z^2 - z^3
-x^3 + y^2*z - 2/3*x*z^2 - 20/27*z^3
-x^3 - x^2*z + y^2*z - x*z^2 - z^3
-x^3 + y^2*z - 54*x*z^2 - 540*z^3
}}}

{{{id=4|
R=Integers(7)
///
}}}

{{{id=5|
x=R(2)
print x, x^2
///
2 4
}}}

{{{id=6|
x.is_square()
///
True
}}}

{{{id=7|
a=x.sqrt()
///
}}}

{{{id=8|
print a, 100*a
///
3 6
}}}

{{{id=9|
print a.lift(), 100*a.lift()
///
3 300
}}}

{{{id=12|
n=random_prime(10^30)*random_prime(10^30)
print n
R=Integers(n)
time R.random_element().is_square()
///
259468041008459855294387276057534250856171095921607899243851
False
Time: CPU 14.05 s, Wall: 14.10 s
}}}

{{{id=13|
p=random_prime(10^100)
print p
R=Integers(p)
time R.random_element().is_square()
///
7639348727956502630901178142032701015745738842489735984849060064941196687282154494289308481814891337
False
Time: CPU 0.11 s, Wall: 0.11 s
}}}

{{{id=19|
next_prime(999312434)
///
999312443
}}}

{{{id=20|

///
}}}