Class groups of number fields obtained by adding p-torsion points of elliptic curves

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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