Number field 4.4.112908.1

• Label: 4.4.112908.1
• Degree: $4$
• Signature: $[4, 0]$
• Discriminant: $112908$
• Root discriminant: $18.33$
• Ramified primes: $2,3,97$
• Class number: $1$
• Class group: trivial
• Galois group: $D_{4}$ (as 4T3)

Normalized defining polynomial

$x^{4} - x^{3} - 16x^{2} - 4x + 16$

Invariants

Degree:		$4$
Signature:		$[4, 0]$
Discriminant:		$112908$ $\medspace = 2^{2}\cdot 3\cdot 97^{2}$
Root discriminant:		$18.33$
Galois root discriminant:		$2\cdot 3^{1/2}97^{1/2}\approx 34.11744421846396$
Ramified primes:		$2$, $3$, $97$
Discriminant root field:		$\Q(\sqrt{3})$
$\Aut(K/\Q)$:		$C_2$
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{3}-\frac{1}{4}a^{2}$

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

Class group and class number

Ideal class group:		Trivial group, which has order $1$
Narrow class group:		$C_{2}$, which has order $2$

Unit group

Rank:		$3$
Torsion generator:		$-1$  (order $2$)
Fundamental units:		$\frac{3}{4}a^{3}-\frac{5}{4}a^{2}-\frac{21}{2}a+3$, $\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{3}{2}a+1$, $\frac{13}{2}a^{3}+\frac{1}{2}a^{2}-99a-113$
Regulator:		$92.4845210605$

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{0}\cdot 92.4845210605 \cdot 1}{2\cdot\sqrt{112908}}\cr\approx \mathstrut & 2.20189538681 \end{aligned}$

Galois group

$D_4$ (as 4T3):

A solvable group of order 8

The 5 conjugacy class representatives for $D_{4}$

Character table for $D_{4}$

Intermediate fields

$\Q(\sqrt{97})$

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

Sibling fields

Galois closure:	8.8.1835743170816.1
Degree 4 sibling:	4.4.13968.1
Minimal sibling:	4.4.13968.1

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	${\href{/padicField/5.4.0.1}{4} }$	${\href{/padicField/7.4.0.1}{4} }$	${\href{/padicField/11.2.0.1}{2} }^{2}$	${\href{/padicField/13.2.0.1}{2} }^{2}$	${\href{/padicField/17.4.0.1}{4} }$	${\href{/padicField/19.4.0.1}{4} }$	${\href{/padicField/23.2.0.1}{2} }^{2}$	${\href{/padicField/29.4.0.1}{4} }$	${\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$	${\href{/padicField/37.2.0.1}{2} }^{2}$	${\href{/padicField/41.4.0.1}{4} }$	${\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$	${\href{/padicField/47.2.0.1}{2} }^{2}$	${\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$	${\href{/padicField/59.2.0.1}{2} }^{2}$

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
$2$	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	2.1.2.2a1.2	$x^{2} + 2 x + 6$	$2$	$1$	$2$	$C_2$	$[2]$
$3$	3.2.1.0a1.1	$x^{2} + 2 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	3.1.2.1a1.1	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$97$	97.2.2.2a1.2	$x^{4} + 192 x^{3} + 9226 x^{2} + 960 x + 122$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$\Q$	$C_1$	$1$	$1$
*	1.97.2t1.a.a	$1$	$97$	$\Q(\sqrt{97})$	$C_2$ (as 2T1)	$1$	$1$
	1.12.2t1.a.a	$1$	$2^{2} \cdot 3$	$\Q(\sqrt{3})$	$C_2$ (as 2T1)	$1$	$1$
	1.1164.2t1.a.a	$1$	$2^{2} \cdot 3 \cdot 97$	$\Q(\sqrt{291})$	$C_2$ (as 2T1)	$1$	$1$
*	2.1164.4t3.a.a	$2$	$2^{2} \cdot 3 \cdot 97$	4.4.112908.1	$D_{4}$ (as 4T3)	$1$	$2$

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