1 | initial version |

Please give us a special case. So far, try `EllipticCurve?`

to see how ot initialize an elliptic curve in sage. There are many examples. $E[p]$ can be "computed" more or less starting for intsance from the corresponding `division_polynomial`

. For instance:

```
p = 3
for a in [3..22]:
print a
E = EllipticCurve( QQ, [-1, a] )
P = E.division_polynomial(3)
L.<a> = NumberField(P)
Cl_L = L.class_group()
if p.divides( Cl_L.order() ):
print "E =", E
print "RANK ::", E.rank()
print "P =", P
print "L =", L
print "Cl(L) ~", L.class_group()
break
```

The above rudimentary search finds:

```
E = Elliptic Curve defined by y^2 = x^3 - x + 19 over Rational Field
RANK :: 1
P = 3*x^4 - 6*x^2 + 228*x - 1
L = Number Field in a with defining polynomial 3*x^4 - 6*x^2 + 228*x - 1
Cl(L) ~ Class group of order 15 with structure C15 of Number Field in a with defining polynomial 3*x^4 - 6*x^2 + 228*x - 1
```

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