Number field 3.1.765415730958295560.7

• Label: 3.1.765415730958295560.7
• Degree: $3$
• Signature: $[1, 1]$
• Discriminant: $-7.654\times 10^{17}$
• Root discriminant: \($914\,743$.07\)
• Ramified primes: $2,3,5,11,13,17,19,23,29,31$
• Class number: not computed
• Class group: not computed
• Galois group: $S_3$ (as 3T2)

Normalized defining polynomial

\( x^{3} + 20036013x - 62873008794 \)

Invariants

Degree:		$3$
Signature:		$[1, 1]$
Discriminant:		\(-765415730958295560\) \(\medspace = -\,2^{3}\cdot 3\cdot 5\cdot 11\cdot 13\cdot 17^{2}\cdot 19^{2}\cdot 23^{2}\cdot 29^{2}\cdot 31^{2}\)
Root discriminant:		\($914\,743$.07\)
Galois root discriminant:		$2^{3/2}3^{1/2}5^{1/2}11^{1/2}13^{1/2}17^{2/3}19^{2/3}23^{2/3}29^{2/3}31^{2/3}\approx 4645709.0609054705$
Ramified primes:		\(2\), \(3\), \(5\), \(11\), \(13\), \(17\), \(19\), \(23\), \(29\), \(31\)
Discriminant root field:		\(\Q(\sqrt{-4290}) \)
$\Aut(K/\Q)$:		$C_1$
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 \(a\))

$1$, $a$, $\frac{1}{426}a^{2}+\frac{7}{142}a-\frac{5}{71}$

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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr = \mathstrut &\frac{2^{1}\cdot(2\pi)^{1}\cdot R \cdot h}{2\cdot\sqrt{765415730958295560}}\cr\mathstrut & \text{ some values not computed } \end{aligned}\]

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	${\href{/padicField/7.3.0.1}{3} }$	R	R	R	R	R	R	R	${\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$	${\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$	${\href{/padicField/43.3.0.1}{3} }$	${\href{/padicField/47.3.0.1}{3} }$	${\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$	${\href{/padicField/59.1.0.1}{1} }^{3}$

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
\(2\)	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	2.1.2.3a1.1	$x^{2} + 2$	$2$	$1$	$3$	$C_2$	$$[3]$$
\(3\)	$\Q_{3}$	$x + 1$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	3.1.2.1a1.2	$x^{2} + 6$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(5\)	$\Q_{5}$	$x + 3$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	5.1.2.1a1.2	$x^{2} + 10$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(11\)	$\Q_{11}$	$x + 9$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	11.1.2.1a1.1	$x^{2} + 11$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(13\)	$\Q_{13}$	$x + 11$	$1$	$1$	$0$	Trivial	$$[\ ]$$
	13.1.2.1a1.2	$x^{2} + 26$	$2$	$1$	$1$	$C_2$	$$[\ ]_{2}$$
\(17\)	17.1.3.2a1.1	$x^{3} + 17$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
\(19\)	19.1.3.2a1.2	$x^{3} + 38$	$3$	$1$	$2$	$C_3$	$$[\ ]_{3}$$
\(23\)	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