Number field 3.1.93964.3

• Label: 3.1.93964.3
• Degree: $3$
• Signature: $[1, 1]$
• Discriminant: $-93964$
• Root discriminant: \(45.46\)
• Ramified primes: $2,13,139$
• Class number: $3$
• Class group: [3]
• Galois group: $S_3$ (as 3T2)

Normalized defining polynomial

\( x^{3} - x^{2} - 69x - 261 \)

Invariants

Degree:		$3$
Signature:		$[1, 1]$
Discriminant:		\(-93964\) \(\medspace = -\,2^{2}\cdot 13^{2}\cdot 139\)
Root discriminant:		\(45.46\)
Galois root discriminant:		$2^{2/3}13^{2/3}139^{1/2}\approx 103.47202902825593$
Ramified primes:		\(2\), \(13\), \(139\)
Discriminant root field:		\(\Q(\sqrt{-139}) \)
$\card{ \Aut(K/\Q) }$:		$1$
This field is not Galois over $\Q$.
This is not a CM field.

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

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

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

Class group and class number

$C_{3}$, which has order $3$

Unit group

Rank:		$1$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental unit:		$4476a^{2}+71538a-1200931$
Regulator:		\( 28.3822115736 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{1}\cdot 28.3822115736 \cdot 3}{2\cdot\sqrt{93964}}\cr\approx \mathstrut & 1.74528601274 \end{aligned}\]

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:	6.0.1227263428144.3
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	R	${\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$	${\href{/padicField/5.1.0.1}{1} }^{3}$	${\href{/padicField/7.3.0.1}{3} }$	${\href{/padicField/11.1.0.1}{1} }^{3}$	R	${\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$	${\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$	${\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$	${\href{/padicField/29.1.0.1}{1} }^{3}$	${\href{/padicField/31.3.0.1}{3} }$	${\href{/padicField/37.3.0.1}{3} }$	${\href{/padicField/41.3.0.1}{3} }$	${\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$	${\href{/padicField/47.3.0.1}{3} }$	${\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$	${\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(2\)	2.3.2.1	$x^{3} + 2$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$
\(13\)	13.3.2.2	$x^{3} + 13$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
\(139\)	$\Q_{139}$	$x + 137$	$1$	$1$	$0$	Trivial	$[\ ]$
	139.2.1.2	$x^{2} + 139$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
	1.139.2t1.a.a	$1$	$ 139 $	\(\Q(\sqrt{-139}) \)	$C_2$ (as 2T1)	$1$	$-1$
*	2.93964.3t2.c.a	$2$	$ 2^{2} \cdot 13^{2} \cdot 139 $	3.1.93964.3	$S_3$ (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 *.