Number field 2.2.21877.1: Q(21877)\Q(\sqrt{21877})

• Label: 2.2.21877.1
• Degree: $2$
• Signature: $[2, 0]$
• Discriminant: $21877$
• Root discriminant: $147.91$
• Ramified primes: $131,167$
• Class number: $3$
• Class group: [3]
• Galois group: $C_2$ (as 2T1)

Normalized defining polynomial

$x^{2} - x - 5469$

Invariants

Degree:		$2$
Signature:		$[2, 0]$
Discriminant:		$21877$ $\medspace = 131\cdot 167$
Root discriminant:		$147.91$
Galois root discriminant:		$131^{1/2}167^{1/2}\approx 147.9087556569928$
Ramified primes:		$131$, $167$
Discriminant root field:		$\Q(\sqrt{21877})$
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:		$C_2$
This field is Galois and abelian over $\Q$.
Conductor:		$21877=131\cdot 167$
Dirichlet character group:		not computed
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator $a$)

$1$, $a$

Monogenic:		Yes
Index:		$1$
Inessential primes:		None

Class group and class number

Ideal class group:		$C_{3}$, which has order $3$
Narrow class group:		$C_{6}$, which has order $6$

Unit group

Rank:		$1$
Torsion generator:		$-1$  (order $2$)
Fundamental unit:		$319125384a+23441156531$
Regulator:		$24.5776902046$

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{0}\cdot 24.5776902046 \cdot 3}{2\cdot\sqrt{21877}}\cr\approx \mathstrut & 0.997007517051 \end{aligned}$

Galois group

$C_2$ (as 2T1):

A cyclic group of order 2

The 2 conjugacy class representatives for $C_2$

Character table for $C_2$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.2.0.1}{2} }$	${\href{/padicField/3.1.0.1}{1} }^{2}$	${\href{/padicField/5.2.0.1}{2} }$	${\href{/padicField/7.1.0.1}{1} }^{2}$	${\href{/padicField/11.1.0.1}{1} }^{2}$	${\href{/padicField/13.2.0.1}{2} }$	${\href{/padicField/17.1.0.1}{1} }^{2}$	${\href{/padicField/19.2.0.1}{2} }$	${\href{/padicField/23.1.0.1}{1} }^{2}$	${\href{/padicField/29.2.0.1}{2} }$	${\href{/padicField/31.2.0.1}{2} }$	${\href{/padicField/37.1.0.1}{1} }^{2}$	${\href{/padicField/41.2.0.1}{2} }$	${\href{/padicField/43.2.0.1}{2} }$	${\href{/padicField/47.2.0.1}{2} }$	${\href{/padicField/53.2.0.1}{2} }$	${\href{/padicField/59.2.0.1}{2} }$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$131$	131.1.2.1a1.2	$x^{2} + 262$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$167$	167.1.2.1a1.1	$x^{2} + 167$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$\Q$	$C_1$	$1$	$1$
*	1.21877.2t1.a.a	$1$	$131 \cdot 167$	$\Q(\sqrt{21877})$	$C_2$ (as 2T1)	$1$	$1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

Spectrum of ring of integers