Global number field 22.22.1275118148086621135238339811277472268288.1

• Label: 22.22.1275118148...8288.1
• Degree: $22$
• Signature: $[22, 0]$
• Discriminant: $2^{22}\cdot 3^{11}\cdot 23^{20}$
• Root discriminant: $59.91$
• Ramified primes: $2, 3, 23$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{22}$ (as 22T1)

Normalized defining polynomial

\( x^{22} - 2 x^{21} - 52 x^{20} + 98 x^{19} + 1099 x^{18} - 1928 x^{17} - 12143 x^{16} + 19510 x^{15} + 75178 x^{14} - 108194 x^{13} - 259836 x^{12} + 324714 x^{11} + 471886 x^{10} - 489902 x^{9} - 393631 x^{8} + 319292 x^{7} + 115208 x^{6} - 68812 x^{5} - 6974 x^{4} + 3640 x^{3} + 132 x^{2} - 48 x + 1 \)

Invariants

Degree:		$22$
Signature:		$[22, 0]$
Discriminant:		\(1275118148086621135238339811277472268288=2^{22}\cdot 3^{11}\cdot 23^{20}\)
Root discriminant:		$59.91$
Ramified primes:		$2, 3, 23$
This field is Galois and abelian over $\Q$.
Conductor:		\(276=2^{2}\cdot 3\cdot 23\)
Dirichlet character group:		$\lbrace$$\chi_{276}(1,·)$, $\chi_{276}(131,·)$, $\chi_{276}(133,·)$, $\chi_{276}(193,·)$, $\chi_{276}(265,·)$, $\chi_{276}(119,·)$, $\chi_{276}(13,·)$, $\chi_{276}(85,·)$, $\chi_{276}(215,·)$, $\chi_{276}(25,·)$, $\chi_{276}(239,·)$, $\chi_{276}(95,·)$, $\chi_{276}(35,·)$, $\chi_{276}(167,·)$, $\chi_{276}(169,·)$, $\chi_{276}(71,·)$, $\chi_{276}(47,·)$, $\chi_{276}(49,·)$, $\chi_{276}(179,·)$, $\chi_{276}(73,·)$, $\chi_{276}(121,·)$, $\chi_{276}(59,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{47} a^{20} + \frac{5}{47} a^{19} + \frac{10}{47} a^{18} + \frac{21}{47} a^{17} + \frac{12}{47} a^{16} - \frac{8}{47} a^{15} + \frac{16}{47} a^{14} - \frac{5}{47} a^{13} - \frac{1}{47} a^{12} - \frac{1}{47} a^{11} - \frac{7}{47} a^{10} + \frac{9}{47} a^{9} + \frac{21}{47} a^{8} - \frac{7}{47} a^{7} - \frac{5}{47} a^{6} - \frac{15}{47} a^{5} + \frac{6}{47} a^{4} + \frac{9}{47} a^{3} + \frac{19}{47} a^{2} + \frac{21}{47} a - \frac{7}{47}$, $\frac{1}{2953897106347955368686850438414567250496707} a^{21} - \frac{9207091858560191727488468658594823095867}{2953897106347955368686850438414567250496707} a^{20} - \frac{696829580382850938294680222124393746825490}{2953897106347955368686850438414567250496707} a^{19} - \frac{111724150302599908334697876451682452454127}{2953897106347955368686850438414567250496707} a^{18} + \frac{711290045932281797510842733249699248048090}{2953897106347955368686850438414567250496707} a^{17} + \frac{502654545150724119028192766251629894487802}{2953897106347955368686850438414567250496707} a^{16} + \frac{170843305712267495047747355273885575906168}{2953897106347955368686850438414567250496707} a^{15} - \frac{1393671321004171222394257186239440583637268}{2953897106347955368686850438414567250496707} a^{14} - \frac{306695090858449549175663771067316802796262}{2953897106347955368686850438414567250496707} a^{13} + \frac{1326904887170545199464477589485274214606622}{2953897106347955368686850438414567250496707} a^{12} - \frac{1157846825625746317300055051957235666037513}{2953897106347955368686850438414567250496707} a^{11} - \frac{1236121577699223932338801232787447256211372}{2953897106347955368686850438414567250496707} a^{10} + \frac{1459672642008319973631904096762187460787116}{2953897106347955368686850438414567250496707} a^{9} - \frac{1252155452051414849898191919466690880982148}{2953897106347955368686850438414567250496707} a^{8} + \frac{1361722812608150388300786178105763908378971}{2953897106347955368686850438414567250496707} a^{7} - \frac{448057830800039434043819277918291315746391}{2953897106347955368686850438414567250496707} a^{6} + \frac{1192255179853116622867642724064102848220096}{2953897106347955368686850438414567250496707} a^{5} + \frac{1440143703973747061316224825471806745495177}{2953897106347955368686850438414567250496707} a^{4} + \frac{13462994285891009282556258718856250102487}{62848874603147986567805328476905686180781} a^{3} - \frac{1096171959342201311201676018038401813334769}{2953897106347955368686850438414567250496707} a^{2} - \frac{727148252153634725435192957952335772774233}{2953897106347955368686850438414567250496707} a - \frac{1394636848876808239892341625642816136408916}{2953897106347955368686850438414567250496707}$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$21$
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:		\( 3259721489230 \) (assuming GRH)

Galois group

$C_{22}$ (as 22T1):

A cyclic group of order 22

The 22 conjugacy class representatives for $C_{22}$

Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\zeta_{23})^+\)

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

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	$22$	$22$	${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$	$22$	$22$	R	$22$	$22$	${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$	$22$	$22$	${\href{/LocalNumberField/47.1.0.1}{1} }^{22}$	$22$	${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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	Data not computed
3	Data not computed
$23$	23.11.10.10	$x^{11} - 23$	$11$	$1$	$10$	$C_{11}$	$[\ ]_{11}$
	23.11.10.10	$x^{11} - 23$	$11$	$1$	$10$	$C_{11}$	$[\ ]_{11}$