Number field 4.0.1600.1: \(\Q(\sqrt{-2}, \sqrt{5})\)

• Label: 4.0.1600.1
• Degree: $4$
• Signature: $[0, 2]$
• Discriminant: $1600$
• Root discriminant: \(6.32\)
• Ramified primes: $2,5$
• Class number: $1$
• Class group: trivial
• Galois group: $C_2^2$ (as 4T2)

Normalized defining polynomial

\( x^{4} + 6x^{2} + 4 \)

Invariants

Degree:		$4$
Signature:		$[0, 2]$
Discriminant:		\(1600\) \(\medspace = 2^{6}\cdot 5^{2}\)
Root discriminant:		\(6.32\)
Galois root discriminant:		$2^{3/2}5^{1/2}\approx 6.324555320336759$
Ramified primes:		\(2\), \(5\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$4$
This field is Galois and abelian over $\Q$.
Conductor:		\(40=2^{3}\cdot 5\)
Dirichlet character group:		$\lbrace$$\chi_{40}(11,·)$, $\chi_{40}(1,·)$, $\chi_{40}(19,·)$, $\chi_{40}(9,·)$$\rbrace$
This is a CM field.
Reflex fields:		\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-10}) \)

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

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

Monogenic:		No
Index:		$1$
Inessential primes:		None

Class group and class number

Trivial group, which has order $1$

Relative class number: $1$

Unit group

Rank:		$1$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental unit:		$\frac{1}{2}a^{2}+1$
Regulator:		\( 0.962423650119 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{2}\cdot 0.962423650119 \cdot 1}{2\cdot\sqrt{1600}}\cr\approx \mathstrut & 0.4749370346463 \end{aligned}\]

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

\(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{5}) \)

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	R	${\href{/padicField/3.2.0.1}{2} }^{2}$	R	${\href{/padicField/7.2.0.1}{2} }^{2}$	${\href{/padicField/11.1.0.1}{1} }^{4}$	${\href{/padicField/13.2.0.1}{2} }^{2}$	${\href{/padicField/17.2.0.1}{2} }^{2}$	${\href{/padicField/19.1.0.1}{1} }^{4}$	${\href{/padicField/23.2.0.1}{2} }^{2}$	${\href{/padicField/29.2.0.1}{2} }^{2}$	${\href{/padicField/31.2.0.1}{2} }^{2}$	${\href{/padicField/37.2.0.1}{2} }^{2}$	${\href{/padicField/41.1.0.1}{1} }^{4}$	${\href{/padicField/43.2.0.1}{2} }^{2}$	${\href{/padicField/47.2.0.1}{2} }^{2}$	${\href{/padicField/53.2.0.1}{2} }^{2}$	${\href{/padicField/59.1.0.1}{1} }^{4}$

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
\(2\)	2.4.6.2	$x^{4} + 4 x^{3} + 16 x^{2} + 24 x + 12$	$2$	$2$	$6$	$C_2^2$	$[3]^{2}$
\(5\)	5.4.2.1	$x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
*	1.40.2t1.b.a	$1$	$ 2^{3} \cdot 5 $	\(\Q(\sqrt{-10}) \)	$C_2$ (as 2T1)	$1$	$-1$
*	1.8.2t1.b.a	$1$	$ 2^{3}$	\(\Q(\sqrt{-2}) \)	$C_2$ (as 2T1)	$1$	$-1$
*	1.5.2t1.a.a	$1$	$ 5 $	\(\Q(\sqrt{5}) \)	$C_2$ (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 *.