Number field 3.1.3618939289355450060.46

• Label: 3.1.361...060.46
• Degree: $3$
• Signature: $[1, 1]$
• Discriminant: $-3.619\times 10^{18}$
• Root discriminant: \($1\,535\,301$.82\)
• Ramified primes: $2,5,7,11,13,17,19,23,29,31$
• Class number: not computed
• Class group: not computed
• Galois group: $S_3$ (as 3T2)

Normalized defining polynomial

\( x^{3} - x^{2} + 18044129x + 131076560441 \)

Invariants

Degree:		$3$
Signature:		$[1, 1]$
Discriminant:		\(-3618939289355450060\) \(\medspace = -\,2^{2}\cdot 5\cdot 7^{2}\cdot 11^{2}\cdot 13\cdot 17^{2}\cdot 19\cdot 23^{2}\cdot 29^{2}\cdot 31^{2}\)
Root discriminant:		\($1\,535\,301$.82\)
Galois root discriminant:		$2^{2/3}5^{1/2}7^{2/3}11^{2/3}13^{1/2}17^{2/3}19^{1/2}23^{2/3}29^{2/3}31^{2/3}\approx 5028883.700476278$
Ramified primes:		\(2\), \(5\), \(7\), \(11\), \(13\), \(17\), \(19\), \(23\), \(29\), \(31\)
Discriminant root field:		\(\Q(\sqrt{-1235}) \)
$\Aut(K/\Q)$:		$C_1$
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 \(a\))

$1$, $a$, $\frac{1}{367}a^{2}-\frac{71}{367}a+\frac{39}{367}$

Monogenic:		Not computed
Index:		$1$
Inessential primes:		None

Class group and class number

Ideal class group:		not computed
Narrow class group:		not computed

Unit group

Rank:		$1$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental unit:		not computed
Regulator:		not computed

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{1}\cdot(2\pi)^{1}\cdot R \cdot h}{2\cdot\sqrt{3618939289355450060}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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:	deg 6
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.3.0.1}{3} }$	R	R	R	R	R	R	R	R	R	${\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$	${\href{/padicField/41.3.0.1}{3} }$	${\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$	${\href{/padicField/47.3.0.1}{3} }$	${\href{/padicField/53.3.0.1}{3} }$	${\href{/padicField/59.3.0.1}{3} }$

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
\(2\)	2.1.3.2a1.1	$x^{3} + 2$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
\(5\)	$\Q_{5}$	$x + 3$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	5.1.2.1a1.2	$x^{2} + 10$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(7\)	7.1.3.2a1.3	$x^{3} + 21$	$3$	$1$	$2$	$C_3$	$$[\ ]_{3}$$
\(11\)	11.1.3.2a1.1	$x^{3} + 11$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
\(13\)	$\Q_{13}$	$x + 11$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	13.1.2.1a1.1	$x^{2} + 13$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(17\)	17.1.3.2a1.1	$x^{3} + 17$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
\(19\)	$\Q_{19}$	$x + 17$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	19.1.2.1a1.2	$x^{2} + 38$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(23\)	23.1.3.2a1.1	$x^{3} + 23$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
\(29\)	29.1.3.2a1.1	$x^{3} + 29$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
\(31\)	31.1.3.2a1.2	$x^{3} + 93$	$3$	$1$	$2$	$C_3$	$$[\ ]_{3}$$

Spectrum of ring of integers