{{{
E = EllipticCurve('37a')
}}}

{{{
def rc():
    return (random(), random(), random())

v = [plot(E.change_ring(GF(p)), pointsize=50, rgbcolor=rc()) \
       for p in primes(60) if p%37]

print len(v)
///
16
}}}

{{{
show(graphics_array(v,4,4), figsize=(6,6), axes=False, frame=True)
}}}

{{{
for p in primes(70):
    print '%4s%4s%4s%6.2f'%(p, E.Np(p), p+1-E.Np(p), float(2*sqrt(p)))
///
   2   5  -2  2.83
   3   7  -3  3.46
   5   8  -2  4.47
   7   9  -1  5.29
  11  17  -5  6.63
  13  16  -2  7.21
  17  18   0  8.25
  19  20   0  8.72
  23  22   2  9.59
  29  24   6 10.77
  31  36  -4 11.14
  37  39  -1 12.17
  41  51  -9 12.81
  43  42   2 13.11
  47  57  -9 13.71
  53  53   1 14.56
  59  52   8 15.36
  61  70  -8 15.62
  67  60   8 16.37
}}}

{{{
P = E(0,0)
}}}

{{{
for n in range(15):
    print n, n*P
///
0 (0 : 1 : 0)
1 (0 : 0 : 1)
2 (1 : 0 : 1)
3 (-1 : -1 : 1)
4 (2 : -3 : 1)
5 (1/4 : -5/8 : 1)
6 (6 : 14 : 1)
7 (-5/9 : 8/27 : 1)
8 (21/25 : -69/125 : 1)
9 (-20/49 : -435/343 : 1)
10 (161/16 : -2065/64 : 1)
11 (116/529 : -3612/12167 : 1)
12 (1357/841 : 28888/24389 : 1)
13 (-3741/3481 : -43355/205379 : 1)
14 (18526/16641 : -2616119/2146689 : 1)
}}}

{{{
def bsdpic(E, pnts=100, xmin=-1, xmax=4):
    print "Rank: ", E.rank()
    print "Generators: ", E.gens()
    Ld = E.Lseries_dokchitser()
    print "L-series", Ld.taylor_series(1,5)
    def L(x): 
       return real(Ld(x))
    show(plot(L,xmin,xmax,plot_points=pnts,plot_division=0,rgbcolor=(0,0,1)),
              xmin=xmin,xmax=xmax)
}}}

{{{
bsdpic(EllipticCurve('37a'))
///
Rank:  1
Generators:  [(0 : 0 : 1)]
L-series 0.305999773834052*z + 0.186547797268162*z^2 + -0.136791463097188*z^3 + O(z^4)
}}}

{{{
# A RANK ZERO CURVE
bsdpic(EllipticCurve('11a'))
///
Rank:  0
Generators:  []
L-series 0.253841860855911 + 0.308708533963172*z + 0.0113280542179202*z^2 + -0.0367068776204353*z^3 + O(z^4)
}}}

{{{
# A RANK TWO CURVE
bsdpic(EllipticCurve('389a'),xmin=-0.1)
///
Rank:  2
Generators:  [(-1 : 1 : 1), (1 : 0 : 1)]
L-series -1.28158145691931e-23 + (7.26268290635587e-24)*z + 0.759316500288427*z^2 + -0.430302337583362*z^3 + O(z^4)
}}}

{{{
# A RANK THREE CURVE
bsdpic(EllipticCurve('5077a'), xmin=0)
///
Rank:  3
Generators:  [(-1 : 3 : 1), (2 : 0 : 1), (4 : 6 : 1)]
L-series (-5.63436295355925e-22)*z + (1.04300238022317e-21)*z^2 + 1.73184990011930*z^3 + O(z^4)
}}}

{{{
# A RANK FOUR CURVE
time bsdpic(EllipticCurve([1, -1, 0, -79, 289]), xmin=0.75, xmax=1.2, pnts=2)
///
Rank:  4
Generators:  [(-10 : 7 : 1), (8 : 7 : 1), (35/4 : 81/8 : 1), (141/16 : 665/64 : 1)]
L-series 1.21908525394358e-24 + (-4.59277879927938e-24)*z + (-4.42853958928154e-22)*z^2 + (1.69062426168686e-21)*z^3 + 8.94384739590089*z^4 + O(z^5)
Time: CPU 0.29 s, Wall: 18.18 s
}}}

{{{

}}}