Number field 4.0.20449.1: \(\Q(\sqrt{-11}, \sqrt{13})\)

• Label: 4.0.20449.1
• Degree: $4$
• Signature: $[0, 2]$
• Discriminant: $20449$
• Root discriminant: \(11.96\)
• Ramified primes: $11,13$
• Class number: $5$
• Class group: [5]
• Galois group: $C_2^2$ (as 4T2)

Normalized defining polynomial

\( x^{4} - x^{2} + 36 \)

Invariants

Degree:		$4$
Signature:		$[0, 2]$
Discriminant:		\(20449\) \(\medspace = 11^{2}\cdot 13^{2}\)
Root discriminant:		\(11.96\)
Galois root discriminant:		$11^{1/2}13^{1/2}\approx 11.958260743101398$
Ramified primes:		\(11\), \(13\)
Discriminant root field:		\(\Q\)
$\card{ \Gal(K/\Q) }$:		$4$
This field is Galois and abelian over $\Q$.
Conductor:		\(143=11\cdot 13\)
Dirichlet character group:		$\lbrace$$\chi_{143}(1,·)$, $\chi_{143}(131,·)$, $\chi_{143}(12,·)$, $\chi_{143}(142,·)$$\rbrace$
This is a CM field.
Reflex fields:		\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-143}) \)

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

$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{12}a^{3}+\frac{5}{12}a-\frac{1}{2}$

Monogenic:		No
Index:		$6$
Inessential primes:		$2$, $3$

Class group and class number

$C_{5}$, which has order $5$

Relative class number: $5$

Unit group

Rank:		$1$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental unit:		$\frac{1}{12}a^{3}-\frac{7}{12}a-\frac{3}{2}$
Regulator:		\( 2.38952643457 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{2}\cdot 2.38952643457 \cdot 5}{2\cdot\sqrt{20449}}\cr\approx \mathstrut & 1.64920843463 \end{aligned}\]

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

\(\Q(\sqrt{13}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-143}) \)

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	${\href{/padicField/2.2.0.1}{2} }^{2}$	${\href{/padicField/3.1.0.1}{1} }^{4}$	${\href{/padicField/5.2.0.1}{2} }^{2}$	${\href{/padicField/7.2.0.1}{2} }^{2}$	R	R	${\href{/padicField/17.2.0.1}{2} }^{2}$	${\href{/padicField/19.2.0.1}{2} }^{2}$	${\href{/padicField/23.1.0.1}{1} }^{4}$	${\href{/padicField/29.2.0.1}{2} }^{2}$	${\href{/padicField/31.2.0.1}{2} }^{2}$	${\href{/padicField/37.2.0.1}{2} }^{2}$	${\href{/padicField/41.2.0.1}{2} }^{2}$	${\href{/padicField/43.2.0.1}{2} }^{2}$	${\href{/padicField/47.2.0.1}{2} }^{2}$	${\href{/padicField/53.1.0.1}{1} }^{4}$	${\href{/padicField/59.2.0.1}{2} }^{2}$

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
\(11\)	11.4.2.1	$x^{4} + 14 x^{3} + 75 x^{2} + 182 x + 620$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
\(13\)	13.4.2.1	$x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
*	1.13.2t1.a.a	$1$	$ 13 $	\(\Q(\sqrt{13}) \)	$C_2$ (as 2T1)	$1$	$1$
*	1.11.2t1.a.a	$1$	$ 11 $	\(\Q(\sqrt{-11}) \)	$C_2$ (as 2T1)	$1$	$-1$
*	1.143.2t1.a.a	$1$	$ 11 \cdot 13 $	\(\Q(\sqrt{-143}) \)	$C_2$ (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 *.