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 \{-3, -4, \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 \{-3, -4, \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
`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`	$\gcd\!\left(a, b\right)$	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), Set(Tuple(a, b, c), For(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(-3)), 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