initial table

K.<a> = NumberField(x^2-x-1) curves=[[1,-a,a,a+2,-a],[1,a-1,a+1,-2*a+3,-1],[1,-1,a+1,-1,-a],[0,-1,1,0,0],[a+1,a-1,a+1,-1,-a+1],[a,a+1,1,1,0]] for c in curves: E = EllipticCurve(K,c) z = E.conductor().gens_reduced()[0] if z[1]: s = str(z[1])+"*a" else: s = "" if z[1] and z[0] > 0: s += "+"+str(z[0]) else: s += str(z[0]) s = "%7s"%(s) s2 = "[" for i in range(5): if c[i] in ZZ: s2 += str(c[i]) else: if c[i][1] == 1: s2 += "a" elif c[i][1] == -1: s2 += "-a" else: s2 += str(c[i][1])+"*a" if c[i][0] > 0: s2 += "+"+str(c[i][0]) elif c[i][0] < 0: s2 += str(c[i][0]) if i < 4: s2 += "," else: s2 += "]" s += "%23s"%(s2) s += "%3s"%(E.simon_two_descent()[1]-E.two_torsion_rank()) s += "%4s"%(E.torsion_order()) z = E.discriminant() B = K.embeddings(RR) if sgn(B[0](z)) < 0: s += " -," else: s += " +," if sgn(B[1](z)) < 0: s += "-" else: s += "+" z = factor(z) s2 = "" for i,k in enumerate(z): s2 += str(k[1]) if i < len(z)-1: s2 += "," s += "%5s"%(s2) z = K(E.j_invariant()) z = z.denominator_ideal() z = factor(z) s2 = "" if not z: s += "0" for i,t in enumerate(z): s2 += str(t[1]) if i < len(z)-1: s2 += "," s += "%5s"%(s2) z = E.tamagawa_numbers() s2 = "" for i,t in enumerate(z): s2 += str(t) if i < len(z)-1: s2 += "," s += "%5s "%(s2) z = E.conductor().factor() for i,t in enumerate(z): s += str(E.kodaira_symbol(t[0])) if i < len(z)-1: s += "," print s 

-2*a+12        [1,-a,a,a+2,-a]  0   7  -,-  7,1  7,1  7,1  I7,I1
 2*a+10  [1,a-1,a+1,-2*a+3,-1]  0   7  -,-  7,1  7,1  7,1  I7,I1
-2*a+12       [1,-1,a+1,-1,-a]  0   5  -,-  1,1  1,1  1,1  I1,I1
     11           [0,-1,1,0,0]  0   5  -,-  1,1  1,1  1,1  I1,I1
 10*a-4  [a+1,a-1,a+1,-1,-a+1]  0   6  +,-  2,1  2,1  2,1  I2,I1
 10*a-6          [a,a+1,1,1,0]  0   6  -,+  2,1  2,1  2,1  I2,I1

-2*a+12        [1,-a,a,a+2,-a]  0   7  -,-  7,1  7,1  7,1  I7,I1
 2*a+10  [1,a-1,a+1,-2*a+3,-1]  0   7  -,-  7,1  7,1  7,1  I7,I1
-2*a+12       [1,-1,a+1,-1,-a]  0   5  -,-  1,1  1,1  1,1  I1,I1
     11           [0,-1,1,0,0]  0   5  -,-  1,1  1,1  1,1  I1,I1
 10*a-4  [a+1,a-1,a+1,-1,-a+1]  0   6  +,-  2,1  2,1  2,1  I2,I1
 10*a-6          [a,a+1,1,1,0]  0   6  -,+  2,1  2,1  2,1  I2,I1

E.two_torsion_rank() 

1

1