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Fungrim entry: dd5681

HD ⁣(x)=(a,b,c)QD(xj ⁣(b+D2a))H_{D}\!\left(x\right) = \prod_{\left(a, b, c\right) \in \mathcal{Q}^{*}_{D}} \left(x - j\!\left(\frac{-b + \sqrt{D}}{2 a}\right)\right)
Assumptions:D{3,4,}andDmod4{0,3}andxCD \in \{-3, -4, \ldots\} \,\mathbin{\operatorname{and}}\, -D \bmod 4 \in \left\{0, 3\right\} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}
TeX:
H_{D}\!\left(x\right) = \prod_{\left(a, b, c\right) \in \mathcal{Q}^{*}_{D}} \left(x - j\!\left(\frac{-b + \sqrt{D}}{2 a}\right)\right)

D \in \{-3, -4, \ldots\} \,\mathbin{\operatorname{and}}\, -D \bmod 4 \in \left\{0, 3\right\} \,\mathbin{\operatorname{and}}\, x \in \mathbb{C}
Definitions:
Fungrim symbol Notation Short description
HilbertClassPolynomialHD ⁣(x)H_{D}\!\left(x\right) Hilbert class polynomial
Productnf(n)\prod_{n} f(n) Product
ModularJj(τ)j(\tau) Modular j-invariant
Sqrtz\sqrt{z} Principal square root
PrimitiveReducedPositiveIntegralBinaryQuadraticFormsQD\mathcal{Q}^{*}_{D} Primitive reduced positive integral binary quadratic forms
ZZLessEqualZn\mathbb{Z}_{\le n} Integers less than or equal to n
CCC\mathbb{C} Complex numbers
Source code for this entry:
Entry(ID("dd5681"),
    Formula(Equal(HilbertClassPolynomial(D, x), Product(Parentheses(Sub(x, ModularJ(Div(Add(Neg(b), Sqrt(D)), Mul(2, a))))), ForElement(Tuple(a, b, c), PrimitiveReducedPositiveIntegralBinaryQuadraticForms(D))))),
    Variables(D, x),
    Assumptions(And(Element(D, ZZLessEqual(-3)), Element(Mod(Neg(D), 4), Set(0, 3)), Element(x, CC))))

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2019-11-19 15:10:20.037976 UTC