Number field 4.0.8649.1: Q(−3,−31)\Q(\sqrt{-3}, \sqrt{-31})

• Label: 4.0.8649.1
• Degree: $4$
• Signature: $[0, 2]$
• Discriminant: $8649$
• Root discriminant: $9.64$
• Ramified primes: $3,31$
• Class number: $3$
• Class group: [3]
• Galois group: $C_2^2$ (as 4T2)

Normalized defining polynomial

$x^{4} - x^{3} - 7x^{2} - 8x + 64$

Invariants

Degree:		$4$
Signature:		$[0, 2]$
Discriminant:		$8649$ $\medspace = 3^{2}\cdot 31^{2}$
Root discriminant:		$9.64$
Galois root discriminant:		$3^{1/2}31^{1/2}\approx 9.643650760992955$
Ramified primes:		$3$, $31$
Discriminant root field:		$\Q$
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:		$C_2^2$
This field is Galois and abelian over $\Q$.
Conductor:		$93=3\cdot 31$
Dirichlet character group:		$\lbrace$$\chi_{93}(32,·)$, $\chi_{93}(1,·)$, $\chi_{93}(92,·)$, $\chi_{93}(61,·)$$\rbrace$
This is a CM field.
Reflex fields:		$\Q(\sqrt{-3})$, $\Q(\sqrt{-31})$

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

$1$, $a$, $a^{2}$, $\frac{1}{56}a^{3}+\frac{1}{8}a^{2}-\frac{1}{8}a-\frac{1}{7}$

Monogenic:		No
Index:		$2$
Inessential primes:		$2$

Class group and class number

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

Unit group

Rank:		$1$
Torsion generator:		$\frac{1}{56} a^{3} + \frac{1}{8} a^{2} - \frac{1}{8} a - \frac{1}{7}$  (order $6$)
Fundamental unit:		$\frac{1}{56}a^{3}+\frac{1}{8}a^{2}-\frac{9}{8}a+\frac{13}{7}$
Regulator:		$3.36610464293$

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{2}\cdot 3.36610464293 \cdot 3}{6\cdot\sqrt{8649}}\cr\approx \mathstrut & 0.714454219320 \end{aligned}$

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

$\Q(\sqrt{-3})$, $\Q(\sqrt{-31})$, $\Q(\sqrt{93})$

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	${\href{/padicField/2.2.0.1}{2} }^{2}$	R	${\href{/padicField/5.2.0.1}{2} }^{2}$	${\href{/padicField/7.1.0.1}{1} }^{4}$	${\href{/padicField/11.2.0.1}{2} }^{2}$	${\href{/padicField/13.2.0.1}{2} }^{2}$	${\href{/padicField/17.2.0.1}{2} }^{2}$	${\href{/padicField/19.1.0.1}{1} }^{4}$	${\href{/padicField/23.2.0.1}{2} }^{2}$	${\href{/padicField/29.2.0.1}{2} }^{2}$	R	${\href{/padicField/37.2.0.1}{2} }^{2}$	${\href{/padicField/41.2.0.1}{2} }^{2}$	${\href{/padicField/43.2.0.1}{2} }^{2}$	${\href{/padicField/47.2.0.1}{2} }^{2}$	${\href{/padicField/53.2.0.1}{2} }^{2}$	${\href{/padicField/59.2.0.1}{2} }^{2}$

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
$3$	3.2.2.2a1.2	$x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$31$	31.1.2.1a1.1	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.1.2.1a1.1	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$\Q$	$C_1$	$1$	$1$
*	1.93.2t1.a.a	$1$	$3 \cdot 31$	$\Q(\sqrt{93})$	$C_2$ (as 2T1)	$1$	$1$
*	1.31.2t1.a.a	$1$	$31$	$\Q(\sqrt{-31})$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3.2t1.a.a	$1$	$3$	$\Q(\sqrt{-3})$	$C_2$ (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 *.

Spectrum of ring of integers