Fungrim entry: dd5681

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 \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
`HilbertClassPolynomial`	$H_{D}\!\left(x\right)$	Hilbert class polynomial
`Product`	$\prod_{n} f(n)$	Product
`ModularJ`	$j(\tau)$	Modular j-invariant
`Sqrt`	$\sqrt{z}$	Principal square root
`PrimitiveReducedPositiveIntegralBinaryQuadraticForms`	$\mathcal{Q}^{*}_{D}$	Primitive reduced positive integral binary quadratic forms
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n
`CC`	$\mathbb{C}$	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))))

Topics using this entry

Modular j-invariant