Number field 3.1.165699.1

• Label: 3.1.165699.1
• Degree: $3$
• Signature: $[1, 1]$
• Discriminant: $-165699$
• Root discriminant: $54.93$
• Ramified primes: $3,17,19$
• Class number: $3$
• Class group: [3]
• Galois group: $S_3$ (as 3T2)

Normalized defining polynomial

$x^{3} - 57x - 228$

Invariants

Degree:		$3$
Signature:		$[1, 1]$
Discriminant:		$-165699$ $\medspace = -\,3^{3}\cdot 17\cdot 19^{2}$
Root discriminant:		$54.93$
Galois root discriminant:		$3^{7/6}17^{1/2}19^{2/3}\approx 105.77141763549201$
Ramified primes:		$3$, $17$, $19$
Discriminant root field:		$\Q(\sqrt{-51})$
$\Aut(K/\Q)$:		$C_1$
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 $a$)

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

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

Class group and class number

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

Unit group

Rank:		$1$
Torsion generator:		$-1$  (order $2$)
Fundamental unit:		$735a^{2}-2373a-38875$
Regulator:		$20.2868409842$

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{1}\cdot 20.2868409842 \cdot 3}{2\cdot\sqrt{165699}}\cr\approx \mathstrut & 0.939410920352 \end{aligned}$

Galois group

$S_3$ (as 3T2):

A solvable group of order 6

The 3 conjugacy class representatives for $S_3$

Character table for $S_3$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Galois closure:	6.0.1400264088651.1
Minimal sibling:	This field is its own minimal sibling

Multiplicative Galois module structure

$U_{K^{gal}}/\textrm{Tors}(U_{K^{gal}}) \cong$ $A'$

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$	R	${\href{/padicField/5.3.0.1}{3} }$	${\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$	${\href{/padicField/11.3.0.1}{3} }$	${\href{/padicField/13.3.0.1}{3} }$	R	R	${\href{/padicField/23.3.0.1}{3} }$	${\href{/padicField/29.3.0.1}{3} }$	${\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$	${\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$	${\href{/padicField/41.3.0.1}{3} }$	${\href{/padicField/43.1.0.1}{1} }^{3}$	${\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$	${\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$	${\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
$3$	3.1.3.3a1.2	$x^{3} + 6 x + 3$	$3$	$1$	$3$	$S_3$	$[\frac{3}{2}]_{2}$
$17$	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$[\ ]$
	17.1.2.1a1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$19$	19.1.3.2a1.1	$x^{3} + 19$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$\Q$	$C_1$	$1$	$1$
	1.51.2t1.a.a	$1$	$3 \cdot 17$	$\Q(\sqrt{-51})$	$C_2$ (as 2T1)	$1$	$-1$
*	2.165699.3t2.b.a	$2$	$3^{3} \cdot 17 \cdot 19^{2}$	3.1.165699.1	$S_3$ (as 3T2)	$1$	$0$

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