Global number field 4.0.441.1: \(\Q(\sqrt{-3}, \sqrt{-7})\)

• Label: 4.0.441.1
• Degree: $4$
• Signature: $[0, 2]$
• Discriminant: $3^{2}\cdot 7^{2}$
• Root discriminant: $4.58$
• Ramified primes: $3, 7$
• Class number: $1$
• Class group: Trivial
• Galois group: $C_2^2$ (as 4T2)

Normalized defining polynomial

\( x^{4} - x^{3} - x^{2} - 2 x + 4 \)

Invariants

Degree:		$4$
Signature:		$[0, 2]$
Discriminant:		\(441=3^{2}\cdot 7^{2}\)
Root discriminant:		$4.58$
Ramified primes:		$3, 7$
$|\Gal(K/\Q)|$:		$4$
This field is Galois and abelian over $\Q$.
Conductor:		\(21=3\cdot 7\)
Dirichlet character group:		$\lbrace$$\chi_{21}(8,·)$, $\chi_{21}(1,·)$, $\chi_{21}(20,·)$, $\chi_{21}(13,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$1$
Torsion generator:		\( -\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{2} a + 2 \) (order $6$)
Fundamental unit:		\( \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{2} a - 1 \)
Regulator:		\( 1.56679923697 \)

Galois group

$C_2^2$ (as 4T2):

An abelian group of order 4

The 4 conjugacy class representatives for $C_2^2$

Character table for $C_2^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Multiplicative Galois module structure

$U_{K^{gal}}/\textrm{Tors}(U_{K^{gal}}) \cong$ $A_1$

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.2.0.1}{2} }^{2}$	R	${\href{/LocalNumberField/5.2.0.1}{2} }^{2}$	R	${\href{/LocalNumberField/11.2.0.1}{2} }^{2}$	${\href{/LocalNumberField/13.2.0.1}{2} }^{2}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{2}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{2}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{2}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{2}$	${\href{/LocalNumberField/37.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{2}$	${\href{/LocalNumberField/43.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{2}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$3$	3.4.2.1	$x^{4} + 9 x^{2} + 36$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$7$	7.2.1.2	$x^{2} + 14$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.2	$x^{2} + 14$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.7.2t1.1c1	$1$	$ 7 $	$x^{2} - x + 2$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3_7.2t1.1c1	$1$	$ 3 \cdot 7 $	$x^{2} - x - 5$	$C_2$ (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 *.