Fungrim entry: 0b4d4b

\mathcal{Q}^{*}_{D} = \left\{ \left(a, b, c\right) : a \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, c \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, {b}^{2} - 4 a c = D \,\mathbin{\operatorname{and}}\, \left|b\right| \le a \le c \,\mathbin{\operatorname{and}}\, \left(\left(\left|b\right| = a \,\mathbin{\operatorname{or}}\, a = c\right) \implies \left(b \ge 0\right)\right) \,\mathbin{\operatorname{and}}\, \gcd\!\left(a, b, c\right) = 1 \right\}

Assumptions:

D \in \{-1, -2, \ldots\} \,\mathbin{\operatorname{and}}\, -D \bmod 4 \in \left\{0, 3\right\}

References:

H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, Definition 5.3.2

TeX:

\mathcal{Q}^{*}_{D} = \left\{ \left(a, b, c\right) : a \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, b \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, c \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, {b}^{2} - 4 a c = D \,\mathbin{\operatorname{and}}\, \left|b\right| \le a \le c \,\mathbin{\operatorname{and}}\, \left(\left(\left|b\right| = a \,\mathbin{\operatorname{or}}\, a = c\right) \implies \left(b \ge 0\right)\right) \,\mathbin{\operatorname{and}}\, \gcd\!\left(a, b, c\right) = 1 \right\}

D \in \{-1, -2, \ldots\} \,\mathbin{\operatorname{and}}\, -D \bmod 4 \in \left\{0, 3\right\}

Definitions:

Fungrim symbol	Notation	Short description
`PrimitiveReducedPositiveIntegralBinaryQuadraticForms`	$\mathcal{Q}^{*}_{D}$	Primitive reduced positive integral binary quadratic forms
`SetBuilder`	$\left\{ f\!\left(x\right) : P\!\left(x\right) \right\}$	Set comprehension
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZ`	$\mathbb{Z}$	Integers
`Pow`	${a}^{b}$	Power
`Abs`	$\left\|z\right\|$	Absolute value
`GCD`	$\gcd\!\left(n, k\right)$	Greatest common divisor
`ZZLessEqual`	$\mathbb{Z}_{\le n}$	Integers less than or equal to n

Source code for this entry:

Entry(ID("0b4d4b"),
    Formula(Equal(PrimitiveReducedPositiveIntegralBinaryQuadraticForms(D), SetBuilder(Tuple(a, b, c), Tuple(a, b, c), And(Element(a, ZZGreaterEqual(1)), Element(b, ZZ), Element(c, ZZ), Equal(Sub(Pow(b, 2), Mul(Mul(4, a), c)), D), LessEqual(Abs(b), a, c), Parentheses(Implies(Or(Equal(Abs(b), a), Equal(a, c)), GreaterEqual(b, 0))), Equal(GCD(a, b, c), 1))))),
    Variables(D),
    Assumptions(And(Element(D, ZZLessEqual(-1)), Element(Mod(Neg(D), 4), Set(0, 3)))),
    References("H. Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, Definition 5.3.2"))

Topics using this entry

Modular j-invariant