Number field 14.0.1521288950440017380097044480.1

• Label: 14.0.152...480.1
• Degree: $14$
• Signature: $[0, 7]$
• Discriminant: $-1.521\times 10^{27}$
• Root discriminant: $87.42$
• Ramified primes: see page
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $S_{14}$ (as 14T63)

Normalized defining polynomial

$x^{14} - 6x + 8$

Invariants

Degree:		$14$
Signature:		$[0, 7]$
Discriminant:		$-1521288950440017380097044480$ $\medspace = -\,2^{12}\cdot 5\cdot 293\cdot 1039\cdot 244004924113840013$
Root discriminant:		$87.42$
Galois root discriminant:		$2^{12/13}5^{1/2}293^{1/2}1039^{1/2}244004924113840013^{1/2}\approx 1155579219962.0247$
Ramified primes:		$2$, $5$, $293$, $1039$, $244004924113840013$
Discriminant root field:		$\Q(\sqrt{-37140\!\cdots\!87755}$)
$\Aut(K/\Q)$:		$C_1$
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 $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2}a^{13}$

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

Class group and class number

Ideal class group:		Trivial group, which has order $1$ (assuming GRH)
Narrow class group:		Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$6$
Torsion generator:		$-1$  (order $2$)
Fundamental units:		$a^{13}-a^{11}-7a^{10}+2a^{8}+27a^{7}+2a^{6}+6a^{5}-33a^{4}-13a^{3}-24a^{2}+38a+9$, $5a^{13}+2a^{12}-7a^{11}-a^{10}+3a^{9}-7a^{8}-14a^{7}-19a^{6}+6a^{5}-25a^{4}-16a^{3}-39a^{2}+29a-63$, $71a^{13}+52a^{12}+55a^{11}+75a^{10}+23a^{9}+65a^{8}+20a^{7}-15a^{6}+54a^{5}-83a^{4}+7a^{3}-38a^{2}-184a-377$, $69a^{13}+925a^{12}-971a^{11}-421a^{10}+1824a^{9}-1168a^{8}-1486a^{7}+3071a^{6}-769a^{5}-3702a^{4}+4631a^{3}+915a^{2}-7569a+5585$, $971a^{13}+1069a^{12}-169a^{11}-1471a^{10}-1319a^{9}+256a^{8}+1833a^{7}+2001a^{6}+175a^{5}-2771a^{4}-3883a^{3}-340a^{2}+5552a+503$, $482a^{13}-715a^{12}+592a^{11}-80a^{10}-720a^{9}+1496a^{8}-1664a^{7}+876a^{6}+753a^{5}-2774a^{4}+4112a^{3}-3516a^{2}+650a+951$ (assuming GRH)
Regulator:		$951832038.935$ (assuming GRH)

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{7}\cdot 951832038.935 \cdot 1}{2\cdot\sqrt{1521288950440017380097044480}}\cr\approx \mathstrut & 4.71719192671 \end{aligned}$ (assuming GRH)

Galois group

$S_{14}$ (as 14T63):

A non-solvable group of order 87178291200

The 135 conjugacy class representatives for $S_{14}$

Character table for $S_{14}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 28 sibling:	data not computed
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	${\href{/padicField/3.6.0.1}{6} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$	R	${\href{/padicField/7.12.0.1}{12} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$	${\href{/padicField/11.7.0.1}{7} }^{2}$	${\href{/padicField/13.12.0.1}{12} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$	${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$	${\href{/padicField/19.14.0.1}{14} }$	${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$	${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$	${\href{/padicField/31.13.0.1}{13} }{,}\,{\href{/padicField/31.1.0.1}{1} }$	${\href{/padicField/37.13.0.1}{13} }{,}\,{\href{/padicField/37.1.0.1}{1} }$	${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$	${\href{/padicField/43.14.0.1}{14} }$	${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$	${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$	${\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
$2$	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	2.1.13.12a1.1	$x^{13} + 2$	$13$	$1$	$12$	$F_{13}$	$[\ ]_{13}^{12}$
$5$	$\Q_{5}$	$x + 3$	$1$	$1$	$0$	Trivial	$[\ ]$
	5.2.1.0a1.1	$x^{2} + 4 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	5.1.2.1a1.2	$x^{2} + 10$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.9.1.0a1.1	$x^{9} + 2 x^{3} + x + 3$	$1$	$9$	$0$	$C_9$	$[\ ]^{9}$
$293$		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $5$	$1$	$5$	$0$	$C_5$	$[\ ]^{5}$
		Deg $5$	$1$	$5$	$0$	$C_5$	$[\ ]^{5}$
$1039$	$\Q_{1039}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $3$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
		Deg $6$	$1$	$6$	$0$	$C_6$	$[\ ]^{6}$
$244004924113840013$		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $12$	$1$	$12$	$0$	$C_{12}$	$[\ ]^{12}$

Spectrum of ring of integers