{{{id=1| K. = QQ[] E = EllipticCurve([1+d-d^2, d^2-d^3, d^2-d^3, 0, 0]) E /// Elliptic Curve defined by y^2 + (-d^2+d+1)*x*y + (-d^3+d^2)*y = x^3 + (-d^3+d^2)*x^2 over Univariate Polynomial Ring in d over Rational Field }}} {{{id=2| E.short_weierstrass_model() /// Elliptic Curve defined by y^2 = x^3 + (-27*d^8+324*d^7-1134*d^6+1512*d^5-945*d^4+378*d^2-108*d-27)*x + (54*d^12-972*d^11+6318*d^10-19116*d^9+30780*d^8-26244*d^7+14742*d^6-11988*d^5+9396*d^4-2484*d^3-810*d^2+324*d+54) over Univariate Polynomial Ring in d over Rational Field }}} {{{id=3| show(E.short_weierstrass_model()) ///
\newcommand{\Bold}[1]{\mathbf{#1}}y^2 = x^3 + \left(-27 d^{8} + 324 d^{7} - 1134 d^{6} + 1512 d^{5} - 945 d^{4} + 378 d^{2} - 108 d - 27\right)x + \left(54 d^{12} - 972 d^{11} + 6318 d^{10} - 19116 d^{9} + 30780 d^{8} - 26244 d^{7} + 14742 d^{6} - 11988 d^{5} + 9396 d^{4} - 2484 d^{3} - 810 d^{2} + 324 d + 54\right)
}}} {{{id=5| /// }}} {{{id=4| for d in [1..1000]: if is_fundamental_discriminant(d): K = QuadraticField(d) if K.class_number() == 5: print "d = ", d break /// d = 401 }}} {{{id=6| for d in [-1,-2,..,-1000]: if is_fundamental_discriminant(d): K = QuadraticField(d) if K.class_number() == 5: print "d = ", d break /// d = -47 }}} {{{id=8| R. = QQ[] for d in [1..500]: f = x^3 + d if not f.is_irreducible(): continue K = NumberField(f, 'a') if K.class_number() == 5: print K break /// Number Field in a with defining polynomial x^3 + 263 }}} {{{id=7| /// }}} {{{id=10| def E(a): return EllipticCurve([0,(a-1),1,-a,0]) for a in [0..80]: print a, E(a).rank() /// 0 0 1 1 2 2 3 2 4 3 5 2 6 2 7 3 8 3 9 3 10 2 11 3 12 3 13 3 14 3 15 2 16 4 17 3 18 2 19 3 20 3 21 2 22 3 23 4 24 3 25 3 26 3 27 2 28 3 29 3 30 2 31 3 32 3 33 2 34 4 35 2 36 3 37 4 38 3 39 2 40 3 41 2 42 4 43 3 44 2 45 2 46 4 47 3 48 3 49 4 50 2 51 2 52 2 53 4 54 4 55 2 56 3 57 3 58 3 59 4 60 3 61 2 62 3 63 3 64 4 65 3 66 3 67 4 68 2 69 3 70 3 71 3 72 3 73 3 74 4 75 3 76 3 77 3 78 3 79 5 80 3 }}} {{{id=11| # so smallest are: 0,1,2,4,16,79. /// }}} {{{id=12| /// }}}