Global number field 22.0.20375776275370821024303050129100543619.1

• Label: 22.0.20375776275...3619.1
• Degree: $22$
• Signature: $[0, 11]$
• Discriminant: $-\,19^{11}\cdot 211^{10}$
• Root discriminant: $49.64$
• Ramified primes: $19, 211$
• Class number: $23$ (GRH)
• Class group: $[23]$ (GRH)
• Galois group: $D_{22}$ (as 22T3)

Normalized defining polynomial

\( x^{22} - 11 x^{21} + 90 x^{20} - 515 x^{19} + 2404 x^{18} - 9153 x^{17} + 29609 x^{16} - 81730 x^{15} + 195276 x^{14} - 404718 x^{13} + 730608 x^{12} - 1147506 x^{11} + 1565057 x^{10} - 1843981 x^{9} + 1862690 x^{8} - 1595475 x^{7} + 1141091 x^{6} - 667142 x^{5} + 309476 x^{4} - 109065 x^{3} + 27307 x^{2} - 4313 x + 323 \)

Invariants

Degree:		$22$
Signature:		$[0, 11]$
Discriminant:		\(-20375776275370821024303050129100543619=-\,19^{11}\cdot 211^{10}\)
Root discriminant:		$49.64$
Ramified primes:		$19, 211$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13} a^{18} + \frac{4}{13} a^{17} + \frac{2}{13} a^{16} + \frac{6}{13} a^{15} - \frac{3}{13} a^{14} - \frac{5}{13} a^{13} - \frac{3}{13} a^{12} + \frac{5}{13} a^{11} + \frac{3}{13} a^{10} + \frac{2}{13} a^{9} - \frac{1}{13} a^{8} - \frac{5}{13} a^{7} - \frac{5}{13} a^{6} + \frac{1}{13} a^{5} - \frac{1}{13} a^{4} - \frac{3}{13} a^{3} + \frac{2}{13} a^{2} - \frac{4}{13}$, $\frac{1}{13} a^{19} - \frac{1}{13} a^{17} - \frac{2}{13} a^{16} - \frac{1}{13} a^{15} - \frac{6}{13} a^{14} + \frac{4}{13} a^{13} + \frac{4}{13} a^{12} - \frac{4}{13} a^{11} + \frac{3}{13} a^{10} + \frac{4}{13} a^{9} - \frac{1}{13} a^{8} + \frac{2}{13} a^{7} - \frac{5}{13} a^{6} - \frac{5}{13} a^{5} + \frac{1}{13} a^{4} + \frac{1}{13} a^{3} + \frac{5}{13} a^{2} - \frac{4}{13} a + \frac{3}{13}$, $\frac{1}{8076887} a^{20} - \frac{10}{8076887} a^{19} - \frac{12228}{351169} a^{18} + \frac{2531481}{8076887} a^{17} + \frac{3278487}{8076887} a^{16} - \frac{2840554}{8076887} a^{15} - \frac{1522233}{8076887} a^{14} + \frac{2742228}{8076887} a^{13} + \frac{3337462}{8076887} a^{12} - \frac{2045200}{8076887} a^{11} + \frac{681015}{8076887} a^{10} - \frac{3607920}{8076887} a^{9} - \frac{32253}{621299} a^{8} - \frac{3450623}{8076887} a^{7} - \frac{1629643}{8076887} a^{6} - \frac{455875}{1153841} a^{5} - \frac{2758699}{8076887} a^{4} - \frac{46906}{8076887} a^{3} + \frac{3416279}{8076887} a^{2} - \frac{324342}{1153841} a + \frac{17677}{36547}$, $\frac{1}{2802679789} a^{21} + \frac{163}{2802679789} a^{20} + \frac{1889922}{215590753} a^{19} - \frac{66626598}{2802679789} a^{18} - \frac{171463894}{400382827} a^{17} + \frac{27276714}{400382827} a^{16} + \frac{56147757}{400382827} a^{15} - \frac{1277049245}{2802679789} a^{14} - \frac{717636370}{2802679789} a^{13} + \frac{6458485}{23551931} a^{12} + \frac{236474166}{2802679789} a^{11} + \frac{272638920}{2802679789} a^{10} + \frac{60717226}{121855643} a^{9} + \frac{28532081}{164863517} a^{8} - \frac{187908783}{2802679789} a^{7} - \frac{323018603}{2802679789} a^{6} + \frac{87220566}{215590753} a^{5} + \frac{593817643}{2802679789} a^{4} - \frac{944102547}{2802679789} a^{3} + \frac{62254386}{215590753} a^{2} - \frac{504433159}{2802679789} a - \frac{42731006}{164863517}$

Class group and class number

$C_{23}$, which has order $23$ (assuming GRH)

Unit group

Rank:		$10$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 294726306.155 \) (assuming GRH)

Galois group

$D_{22}$ (as 22T3):

A solvable group of order 44

The 14 conjugacy class representatives for $D_{22}$

Character table for $D_{22}$

Intermediate fields

\(\Q(\sqrt{-19}) \), 11.11.1035571956771279049.1

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

Sibling fields

Galois closure:	data not computed
Degree 22 sibling:	data not computed

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	$22$	$22$	${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/7.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/13.2.0.1}{2} }^{11}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$	R	${\href{/LocalNumberField/23.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$	$22$	$22$	${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$	$22$	${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{11}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{11}$

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
$19$	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
211	Data not computed