Number field 3.1.23.1

• Label: 3.1.23.1
• Degree: $3$
• Signature: $[1, 1]$
• Discriminant: $-23$
• Root discriminant: \(2.84\)
• Ramified prime: $23$
• Class number: $1$
• Class group: trivial
• Galois group: $S_3$ (as 3T2)

This is the cubic field with the smallest absolute discriminant.

Normalized defining polynomial

\( x^{3} - x^{2} + 1 \)

Invariants

Degree:		$3$
Signature:		$[1, 1]$
Discriminant:		\(-23\)
Root discriminant:		\(2.84\)
Galois root discriminant:		$23^{1/2}\approx 4.795831523312719$
Ramified primes:		\(23\)
Discriminant root field:		\(\Q(\sqrt{-23}) \)
$\card{ \Aut(K/\Q) }$:		$1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$

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

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$1$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental unit:		$a^{2}-a$
Regulator:		\( 0.281199574323 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{1}\cdot 0.281199574323 \cdot 1}{2\cdot\sqrt{23}}\cr\approx \mathstrut & 0.3684093207159 \end{aligned}\]

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:	6.0.12167.1
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	${\href{/padicField/2.3.0.1}{3} }$	${\href{/padicField/3.3.0.1}{3} }$	${\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$	${\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$	${\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$	${\href{/padicField/13.3.0.1}{3} }$	${\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$	${\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$	R	${\href{/padicField/29.3.0.1}{3} }$	${\href{/padicField/31.3.0.1}{3} }$	${\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$	${\href{/padicField/41.3.0.1}{3} }$	${\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$	${\href{/padicField/47.3.0.1}{3} }$	${\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$	${\href{/padicField/59.1.0.1}{1} }^{3}$

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
\(23\)	$\Q_{23}$	$x + 18$	$1$	$1$	$0$	Trivial	$[\ ]$
	23.2.1.2	$x^{2} + 23$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
	1.23.2t1.a.a	$1$	$ 23 $	\(\Q(\sqrt{-23}) \)	$C_2$ (as 2T1)	$1$	$-1$
*	2.23.3t2.b.a	$2$	$ 23 $	3.1.23.1	$S_3$ (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 *.