Number field 6.0.17904554058816.3

• Label: 6.0.17904554058816.3
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-1.790\times 10^{13}$
• Root discriminant: $161.74$
• Ramified primes: $2,3,19,43$
• Class number: $108$
• Class group: [6, 18]
• Galois group: $S_3$ (as 6T2)

Normalized defining polynomial

$x^{6} + 81x^{4} + 12960x^{2} + 297216$

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		$-17904554058816$ $\medspace = -\,2^{6}\cdot 3^{3}\cdot 19^{4}\cdot 43^{3}$
Root discriminant:		$161.74$
Galois root discriminant:		$2\cdot 3^{1/2}19^{2/3}43^{1/2}\approx 161.74365447852955$
Ramified primes:		$2$, $3$, $19$, $43$
Discriminant root field:		$\Q(\sqrt{-129})$
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:		$S_3$
This field is Galois over $\Q$.
This is not a CM field.
Maximal CM subfield:		$\Q(\sqrt{-129})$

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

$1$, $a$, $\frac{1}{3}a^{2}$, $\frac{1}{36}a^{3}-\frac{5}{12}a$, $\frac{1}{3456}a^{4}-\frac{1}{72}a^{3}-\frac{85}{1152}a^{2}+\frac{5}{24}a-\frac{11}{24}$, $\frac{1}{13824}a^{5}+\frac{43}{4608}a^{3}-\frac{1}{6}a^{2}-\frac{9}{32}a-\frac{1}{2}$

Monogenic:		No
Index:		$8$
Inessential primes:		$2$

Class group and class number

Ideal class group:		$C_{6}\times C_{18}$, which has order $108$
Narrow class group:		$C_{6}\times C_{18}$, which has order $108$

Unit group

Rank:		$2$
Torsion generator:		$-1$  (order $2$)
Fundamental units:		$\frac{40964387949}{128}a^{5}-\frac{515594801729}{432}a^{4}+\frac{2543965591303}{384}a^{3}+\frac{39643626166325}{144}a^{2}-\frac{333843309099}{8}a+\frac{23653467680086}{3}$, $\frac{40972006553}{216}a^{5}-\frac{1859198607179}{1728}a^{4}-\frac{1339297808971}{72}a^{3}-\frac{230781104071705}{576}a^{2}-\frac{2420403587207}{4}a-\frac{115293628568507}{12}$
Regulator:		$4082.63150081$

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 4082.63150081 \cdot 108}{2\cdot\sqrt{17904554058816}}\cr\approx \mathstrut & 12.9238465472 \end{aligned}$

Galois group

$S_3$ (as 6T2):

A solvable group of order 6

The 3 conjugacy class representatives for $S_3$

Character table for $S_3$

Intermediate fields

$\Q(\sqrt{-129})$, 3.1.186276.1 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling algebras

Twin sextic algebra:	3.1.186276.1 $\times$ $\Q$ $\times$ $\Q$ $\times$ $\Q$
Degree 3 sibling:	3.1.186276.1
Minimal sibling:	3.1.186276.1

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	${\href{/padicField/5.3.0.1}{3} }^{2}$	${\href{/padicField/7.3.0.1}{3} }^{2}$	${\href{/padicField/11.1.0.1}{1} }^{6}$	${\href{/padicField/13.1.0.1}{1} }^{6}$	${\href{/padicField/17.2.0.1}{2} }^{3}$	R	${\href{/padicField/23.3.0.1}{3} }^{2}$	${\href{/padicField/29.3.0.1}{3} }^{2}$	${\href{/padicField/31.2.0.1}{2} }^{3}$	${\href{/padicField/37.2.0.1}{2} }^{3}$	${\href{/padicField/41.2.0.1}{2} }^{3}$	R	${\href{/padicField/47.1.0.1}{1} }^{6}$	${\href{/padicField/53.2.0.1}{2} }^{3}$	${\href{/padicField/59.3.0.1}{3} }^{2}$

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
$2$	2.1.2.2a1.1	$x^{2} + 2 x + 2$	$2$	$1$	$2$	$C_2$	$[2]$
	2.1.2.2a1.1	$x^{2} + 2 x + 2$	$2$	$1$	$2$	$C_2$	$[2]$
	2.1.2.2a1.1	$x^{2} + 2 x + 2$	$2$	$1$	$2$	$C_2$	$[2]$
$3$	3.1.2.1a1.1	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.1.2.1a1.1	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.1.2.1a1.1	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$19$	19.1.3.2a1.1	$x^{3} + 19$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	19.1.3.2a1.1	$x^{3} + 19$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
$43$	43.1.2.1a1.2	$x^{2} + 129$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	43.1.2.1a1.2	$x^{2} + 129$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	43.1.2.1a1.2	$x^{2} + 129$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Spectrum of ring of integers