Number field 2.0.1999.1: Q(−1999)\Q(\sqrt{-1999})

• Label: 2.0.1999.1
• Degree: $2$
• Signature: $[0, 1]$
• Discriminant: $-1999$
• Root discriminant: $44.71$
• Ramified prime: $1999$
• Class number: $27$
• Class group: [27]
• Galois group: $C_2$ (as 2T1)

Normalized defining polynomial

$x^{2} - x + 500$

Invariants

Degree:		$2$
Signature:		$[0, 1]$
Discriminant:		$-1999$
Root discriminant:		$44.71$
Galois root discriminant:		$1999^{1/2}\approx 44.710177812216315$
Ramified primes:		$1999$
Discriminant root field:		$\Q(\sqrt{-1999})$
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:		$C_2$
This field is Galois and abelian over $\Q$.
Conductor:		$1999$
Dirichlet character group:		$\lbrace$$\chi_{1999}(1,·)$, $\chi_{1999}(1998,·)$$\rbrace$
This is a CM field.
Reflex fields:		$\Q(\sqrt{-1999})$

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

$1$, $a$

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

Class group and class number

Ideal class group:		$C_{27}$, which has order $27$
Narrow class group:		$C_{27}$, which has order $27$
Relative class number:		$27$

Unit group

Rank:		$0$
Torsion generator:		$-1$  (order $2$)
Regulator:		$1$

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr =\mathstrut &\frac{2^{0}\cdot(2\pi)^{1}\cdot 1 \cdot 27}{2\cdot\sqrt{1999}}\cr\approx \mathstrut & 1.89717433026509 \end{aligned}$

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	${\href{/padicField/2.1.0.1}{1} }^{2}$	${\href{/padicField/3.2.0.1}{2} }$	${\href{/padicField/5.1.0.1}{1} }^{2}$	${\href{/padicField/7.2.0.1}{2} }$	${\href{/padicField/11.1.0.1}{1} }^{2}$	${\href{/padicField/13.1.0.1}{1} }^{2}$	${\href{/padicField/17.2.0.1}{2} }$	${\href{/padicField/19.2.0.1}{2} }$	${\href{/padicField/23.1.0.1}{1} }^{2}$	${\href{/padicField/29.2.0.1}{2} }$	${\href{/padicField/31.1.0.1}{1} }^{2}$	${\href{/padicField/37.1.0.1}{1} }^{2}$	${\href{/padicField/41.1.0.1}{1} }^{2}$	${\href{/padicField/43.2.0.1}{2} }$	${\href{/padicField/47.2.0.1}{2} }$	${\href{/padicField/53.1.0.1}{1} }^{2}$	${\href{/padicField/59.1.0.1}{1} }^{2}$

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
$1999$		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$\Q$	$C_1$	$1$	$1$
*	1.1999.2t1.a.a	$1$	$1999$	$\Q(\sqrt{-1999})$	$C_2$ (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