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NumberFieldClassNumber[θ]

gives the class number for the algebraic number field generated by theta.

Examples  
Basic Examples  
Scope  
Applications  
Properties & Relations  
Possible Issues  
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NumberFieldClassNumber

NumberFieldClassNumber[θ]

gives the class number for the algebraic number field generated by theta.

Examples

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Basic Examples  (1)

Find the class number of the algebraic number field :

Wolfram Language code: NumberFieldClassNumber[Sqrt[5]]

Scope  (2)

Class number of complex and real quadratic fields:

Wolfram Language code: NumberFieldClassNumber[Sqrt[-151]]
Wolfram Language code: NumberFieldClassNumber[Sqrt[3263123]]

NumberFieldClassNumber automatically threads over lists:

Wolfram Language code: NumberFieldClassNumber[{Sqrt[-71], 2 ^ (1 / 3)}]

Applications  (1)

Value of the Dirichlet L-functions at for the real primitive character modulo :

Wolfram Language code: Pi NumberFieldClassNumber[Sqrt[-11]] / Sqrt[11]

Properties & Relations  (1)

Number of representations of 19 as a sum of 3 squares:

Wolfram Language code: 12(1 - JacobiSymbol[-19, 2]) NumberFieldClassNumber[Sqrt[-19]]
Wolfram Language code: SquaresR[3, 19]

Possible Issues  (1)

Class number for some algebraic number fields may not be yet available:

Wolfram Language code: NumberFieldClassNumber[Sqrt[1 + Sqrt[2]]]
Wolfram Research (2007), NumberFieldClassNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/NumberFieldClassNumber.html.

Text

Wolfram Research (2007), NumberFieldClassNumber, Wolfram Language function, https://reference.wolfram.com/language/ref/NumberFieldClassNumber.html.

CMS

Wolfram Language. 2007. "NumberFieldClassNumber." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/NumberFieldClassNumber.html.

APA

Wolfram Language. (2007). NumberFieldClassNumber. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/NumberFieldClassNumber.html

BibTeX

@misc{reference.wolfram_2026_numberfieldclassnumber, author="Wolfram Research", title="{NumberFieldClassNumber}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/NumberFieldClassNumber.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_numberfieldclassnumber, organization={Wolfram Research}, title={NumberFieldClassNumber}, year={2007}, url={https://reference.wolfram.com/language/ref/NumberFieldClassNumber.html}, note=[Accessed: 13-June-2026]}