Number field 2.2.12.1: \(\Q(\sqrt{3}) \)

• Label: 2.2.12.1
• Degree: $2$
• Signature: $[2, 0]$
• Discriminant: $12$
• Root discriminant: \(3.46\)
• Ramified primes: $2,3$
• Class number: $1$
• Class group: trivial
• Galois group: $C_2$ (as 2T1)

Normalized defining polynomial

\( x^{2} - 3 \)

Invariants

Degree:		$2$
Signature:		$[2, 0]$
Discriminant:		\(12\) \(\medspace = 2^{2}\cdot 3\)
Root discriminant:		\(3.46\)
Galois root discriminant:		$2\cdot 3^{1/2}\approx 3.4641016151377544$
Ramified primes:		\(2\), \(3\)
Discriminant root field:		\(\Q(\sqrt{3}) \)
$\card{ \Gal(K/\Q) }$:		$2$
This field is Galois and abelian over $\Q$.
Conductor:		\(12=2^{2}\cdot 3\)
Dirichlet character group:		$\lbrace$$\chi_{12}(1,·)$, $\chi_{12}(11,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$

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

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$1$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental unit:		$a-2$
Regulator:		\( 1.31695789692 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{0}\cdot 1.31695789692 \cdot 1}{2\cdot\sqrt{12}}\cr\approx \mathstrut & 0.760345996298 \end{aligned}\]

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	R	R	${\href{/padicField/5.2.0.1}{2} }$	${\href{/padicField/7.2.0.1}{2} }$	${\href{/padicField/11.1.0.1}{1} }^{2}$	${\href{/padicField/13.1.0.1}{1} }^{2}$	${\href{/padicField/17.2.0.1}{2} }$	${\href{/padicField/19.2.0.1}{2} }$	${\href{/padicField/23.1.0.1}{1} }^{2}$	${\href{/padicField/29.2.0.1}{2} }$	${\href{/padicField/31.2.0.1}{2} }$	${\href{/padicField/37.1.0.1}{1} }^{2}$	${\href{/padicField/41.2.0.1}{2} }$	${\href{/padicField/43.2.0.1}{2} }$	${\href{/padicField/47.1.0.1}{1} }^{2}$	${\href{/padicField/53.2.0.1}{2} }$	${\href{/padicField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. 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
\(2\)	2.2.2.2	$x^{2} + 2 x + 6$	$2$	$1$	$2$	$C_2$	$[2]$
\(3\)	3.2.1.1	$x^{2} + 6$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
*	1.12.2t1.a.a	$1$	$ 2^{2} \cdot 3 $	\(\Q(\sqrt{3}) \)	$C_2$ (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 *.