Number field 3.1.417555469216461876300.37

• Label: 3.1.417...300.37
• Degree: $3$
• Signature: $[1, 1]$
• Discriminant: $-4.176\times 10^{20}$
• Root discriminant: \($7\,474\,314$.92\)
• Ramified primes: $2,3,5,7,11,13,19,23,29,31$
• Class number: $82563624$ (GRH)
• Class group: [3, 3, 3, 3, 3, 3, 6, 18876] (GRH)
• Galois group: $S_3$ (as 3T2)

Normalized defining polynomial

\( x^{3} - 81313514792510 \)

Invariants

Degree:		$3$
Signature:		$[1, 1]$
Discriminant:		\(-417555469216461876300\) \(\medspace = -\,2^{2}\cdot 3^{3}\cdot 5^{2}\cdot 7^{2}\cdot 11^{2}\cdot 13^{2}\cdot 19^{2}\cdot 23^{2}\cdot 29^{2}\cdot 31^{2}\)
Root discriminant:		\($7\,474\,314$.92\)
Galois root discriminant:		$2^{2/3}3^{7/6}5^{2/3}7^{2/3}11^{2/3}13^{2/3}19^{2/3}23^{2/3}29^{2/3}31^{2/3}\approx 8976180.999446508$
Ramified primes:		\(2\), \(3\), \(5\), \(7\), \(11\), \(13\), \(19\), \(23\), \(29\), \(31\)
Discriminant root field:		\(\Q(\sqrt{-3}) \)
$\Aut(K/\Q)$:		$C_1$
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 \(a\))

$1$, $a$, $\frac{1}{20677}a^{2}$

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

Class group and class number

Ideal class group:		$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{6}\times C_{18876}$, which has order $82563624$ (assuming GRH)
Narrow class group:		$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{6}\times C_{18876}$, which has order $82563624$ (assuming GRH)

Unit group

Rank:		$1$
Torsion generator:		\( -1 \)  (order $2$)
Fundamental unit:		$\frac{2646}{20677}a^{2}-5544a-1$ (assuming GRH)
Regulator:		\( 39.692444276089226 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{1}\cdot 39.692444276089226 \cdot 82563624}{2\cdot\sqrt{417555469216461876300}}\cr\approx \mathstrut & 1.00767241800391 \end{aligned}\] (assuming GRH)

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	${\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$	R	R	R	R	${\href{/padicField/37.1.0.1}{1} }^{3}$	${\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$	${\href{/padicField/43.3.0.1}{3} }$	${\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$	${\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$	${\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(2\)	2.1.3.2a1.1	$x^{3} + 2$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
\(3\)	3.1.3.3a1.1	$x^{3} + 3 x + 3$	$3$	$1$	$3$	$S_3$	$$[\frac{3}{2}]_{2}$$
\(5\)	5.1.3.2a1.1	$x^{3} + 5$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
\(7\)	7.1.3.2a1.3	$x^{3} + 21$	$3$	$1$	$2$	$C_3$	$$[\ ]_{3}$$
\(11\)	11.1.3.2a1.1	$x^{3} + 11$	$3$	$1$	$2$	$S_3$	$$[\ ]_{3}^{2}$$
\(13\)	13.1.3.2a1.2	$x^{3} + 26$	$3$	$1$	$2$	$C_3$	$$[\ ]_{3}$$
\(19\)	19.1.3.2a1.1	$x^{3} + 19$	$3$	$1$	$2$	$C_3$	$$[\ ]_{3}$$
\(23\)	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