Number field 3.1.16440344496936360.9

• Label: 3.1.16440344496936360.9
• Degree: $3$
• Signature: $[1, 1]$
• Discriminant: $-1.644\times 10^{16}$
• Root discriminant: \($254\,274$.99\)
• Ramified primes: $2,3,5,7,11,13,17,19,23,29,31$
• Class number: not computed
• Class group: not computed
• Galois group: $S_3$ (as 3T2)

Normalized defining polynomial

\( x^{3} + 19147293x - 27669516534 \)

Invariants

Degree:		$3$
Signature:		$[1, 1]$
Discriminant:		\(-16440344496936360\) \(\medspace = -\,2^{3}\cdot 3^{5}\cdot 5\cdot 7\cdot 11^{2}\cdot 13\cdot 17\cdot 19\cdot 23^{2}\cdot 29\cdot 31\)
Root discriminant:		\($254\,274$.99\)
Galois root discriminant:		$2^{3/2}3^{11/6}5^{1/2}7^{1/2}11^{2/3}13^{1/2}17^{1/2}19^{1/2}23^{2/3}29^{1/2}31^{1/2}\approx 9746180.417869182$
Ramified primes:		\(2\), \(3\), \(5\), \(7\), \(11\), \(13\), \(17\), \(19\), \(23\), \(29\), \(31\)
Discriminant root field:		\(\Q(\sqrt{-792729210}) \)
$\Aut(K/\Q)$:		$C_1$
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 \(a\))

$1$, $a$, $\frac{1}{1722}a^{2}+\frac{51}{574}a-\frac{54}{287}$

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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{1}\cdot(2\pi)^{1}\cdot R \cdot h}{2\cdot\sqrt{16440344496936360}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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	R	R	R	R	R	R	R	R	R	R	${\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$	${\href{/padicField/41.1.0.1}{1} }^{3}$	${\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$	${\href{/padicField/47.3.0.1}{3} }$	${\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$	${\href{/padicField/59.3.0.1}{3} }$

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
\(2\)	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	2.1.2.3a1.2	$x^{2} + 10$	$2$	$1$	$3$	$C_2$	$$[3]$$
\(3\)	3.1.3.5a1.3	$x^{3} + 18 x + 3$	$3$	$1$	$5$	$S_3$	$$[\frac{5}{2}]_{2}$$
\(5\)	$\Q_{5}$	$x + 3$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	5.1.2.1a1.2	$x^{2} + 10$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(7\)	$\Q_{7}$	$x + 4$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	7.1.2.1a1.1	$x^{2} + 7$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(11\)	11.1.3.2a1.1	$x^{3} + 11$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
\(13\)	$\Q_{13}$	$x + 11$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	13.1.2.1a1.2	$x^{2} + 26$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(17\)	$\Q_{17}$	$x + 14$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	17.1.2.1a1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(19\)	$\Q_{19}$	$x + 17$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	19.1.2.1a1.2	$x^{2} + 38$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(23\)	23.1.3.2a1.1	$x^{3} + 23$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
\(29\)	$\Q_{29}$	$x + 27$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	29.1.2.1a1.2	$x^{2} + 58$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(31\)	$\Q_{31}$	$x + 28$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	31.1.2.1a1.1	$x^{2} + 31$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$

Spectrum of ring of integers