Number field 14.2.6928793237157103918561273965.1

• Label: 14.2.692...965.1
• Degree: $14$
• Signature: $[2, 6]$
• Discriminant: $6.929\times 10^{27}$
• Root discriminant: $97.41$
• Ramified primes: $3,5,19,31,255053,5785828481723$
• Class number: $1$ (GRH)
• Class group: trivial (GRH)
• Galois group: $S_{14}$ (as 14T63)

Normalized defining polynomial

$x^{14} - 9x + 3$

Invariants

Degree:		$14$
Signature:		$[2, 6]$
Discriminant:		$6928793237157103918561273965$ $\medspace = 3^{13}\cdot 5\cdot 19\cdot 31\cdot 255053\cdot 5785828481723$
Root discriminant:		$97.41$
Galois root discriminant:		$3^{13/14}5^{1/2}19^{1/2}31^{1/2}255053^{1/2}5785828481723^{1/2}\approx 182844461715.1342$
Ramified primes:		$3$, $5$, $19$, $31$, $255053$, $5785828481723$
Discriminant root field:		$\Q(\sqrt{13037\!\cdots\!78365}$)
$\Aut(K/\Q)$:		$C_1$
This field is not Galois over $\Q$.
This is not a CM field.

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

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

Monogenic:		Yes
Index:		$1$
Inessential primes:		None

Class group and class number

Ideal class group:		Trivial group, which has order $1$ (assuming GRH)
Narrow class group:		$C_{2}$, which has order $2$ (assuming GRH)

Unit group

Rank:		$7$
Torsion generator:		$-1$  (order $2$)
Fundamental units:		$3a^{13}+4a^{12}+5a^{10}+6a^{9}-6a^{8}-5a^{7}+2a^{6}-16a^{5}-17a^{4}+10a^{3}-6a^{2}-12a+13$, $9a^{13}+2a^{12}-16a^{11}+12a^{10}+16a^{9}-25a^{8}+6a^{7}+34a^{6}-38a^{5}-14a^{4}+74a^{3}-31a^{2}-60a+29$, $30a^{13}+36a^{12}+41a^{11}+49a^{10}+50a^{9}+64a^{8}+65a^{7}+91a^{6}+91a^{5}+122a^{4}+116a^{3}+150a^{2}+156a-70$, $14a^{13}-7a^{12}+3a^{11}+a^{10}-15a^{9}+36a^{8}-50a^{7}+64a^{6}-97a^{5}+128a^{4}-136a^{3}+144a^{2}-171a+43$, $94a^{13}+24a^{12}+15a^{11}-2a^{10}+7a^{9}-3a^{8}-2a^{7}+8a^{6}-16a^{5}+28a^{4}-40a^{3}+47a^{2}-51a-787$, $6a^{13}-21a^{12}+32a^{11}-21a^{10}+a^{9}+39a^{8}-59a^{7}+64a^{6}-13a^{5}-53a^{4}+139a^{3}-156a^{2}+114a-8$, $39a^{13}-22a^{12}+14a^{11}+63a^{10}+7a^{9}-84a^{8}-34a^{7}+99a^{6}+55a^{5}-144a^{4}-126a^{3}+166a^{2}+222a-511$ (assuming GRH)
Regulator:		$586657278.452$ (assuming GRH)

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{6}\cdot 586657278.452 \cdot 1}{2\cdot\sqrt{6928793237157103918561273965}}\cr\approx \mathstrut & 0.867291056845 \end{aligned}$ (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	${\href{/padicField/2.7.0.1}{7} }{,}\,{\href{/padicField/2.5.0.1}{5} }{,}\,{\href{/padicField/2.2.0.1}{2} }$	R	R	${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.3.0.1}{3} }$	${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$	${\href{/padicField/13.3.0.1}{3} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$	${\href{/padicField/17.5.0.1}{5} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$	R	${\href{/padicField/23.9.0.1}{9} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$	${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$	R	${\href{/padicField/37.14.0.1}{14} }$	${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$	${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }$	${\href{/padicField/47.6.0.1}{6} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$	${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }^{2}$	${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$

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
$3$	3.1.14.13a1.1	$x^{14} + 3$	$14$	$1$	$13$	$F_7 \times C_2$	$[\ ]_{14}^{6}$
$5$	$\Q_{5}$	$x + 3$	$1$	$1$	$0$	Trivial	$[\ ]$
	5.1.2.1a1.1	$x^{2} + 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.0a1.1	$x^{2} + 4 x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	5.9.1.0a1.1	$x^{9} + 2 x^{3} + x + 3$	$1$	$9$	$0$	$C_9$	$[\ ]^{9}$
$19$	19.1.2.1a1.1	$x^{2} + 19$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.3.1.0a1.1	$x^{3} + 4 x + 17$	$1$	$3$	$0$	$C_3$	$[\ ]^{3}$
	19.9.1.0a1.1	$x^{9} + 11 x^{3} + 14 x^{2} + 16 x + 17$	$1$	$9$	$0$	$C_9$	$[\ ]^{9}$
$31$	31.1.2.1a1.2	$x^{2} + 93$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.2.1.0a1.1	$x^{2} + 29 x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	31.10.1.0a1.1	$x^{10} + 30 x^{5} + 26 x^{4} + 13 x^{3} + 13 x^{2} + 13 x + 3$	$1$	$10$	$0$	$C_{10}$	$[\ ]^{10}$
$255053$		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $4$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
		Deg $8$	$1$	$8$	$0$	$C_8$	$[\ ]^{8}$
$5785828481723$	$\Q_{5785828481723}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{5785828481723}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{5785828481723}$	$x$	$1$	$1$	$0$	Trivial	$[\ ]$
		Deg $2$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
		Deg $9$	$1$	$9$	$0$	$C_9$	$[\ ]^{9}$

Spectrum of ring of integers