Global number field 22.22.4299288794103243236127943577240214703609.1

• Label: 22.22.4299288794...3609.1
• Degree: $22$
• Signature: $[22, 0]$
• Discriminant: $19^{11}\cdot 211^{11}$
• Root discriminant: $63.32$
• Ramified primes: $19, 211$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $D_{11}$ (as 22T2)

Normalized defining polynomial

\( x^{22} - 10 x^{21} - 29 x^{20} + 529 x^{19} - 69 x^{18} - 11577 x^{17} + 11850 x^{16} + 136774 x^{15} - 182767 x^{14} - 954992 x^{13} + 1314121 x^{12} + 4053245 x^{11} - 5095548 x^{10} - 10389210 x^{9} + 10902904 x^{8} + 15714269 x^{7} - 12265214 x^{6} - 13423268 x^{5} + 6295986 x^{4} + 5702075 x^{3} - 940712 x^{2} - 757690 x - 2957 \)

Invariants

Degree:		$22$
Signature:		$[22, 0]$
Discriminant:		\(4299288794103243236127943577240214703609=19^{11}\cdot 211^{11}\)
Root discriminant:		$63.32$
Ramified primes:		$19, 211$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{19} a^{16} - \frac{8}{19} a^{15} + \frac{3}{19} a^{14} + \frac{6}{19} a^{13} - \frac{3}{19} a^{12} - \frac{5}{19} a^{11} - \frac{3}{19} a^{10} + \frac{6}{19} a^{9} + \frac{4}{19} a^{8} - \frac{5}{19} a^{7} + \frac{5}{19} a^{6} + \frac{5}{19} a^{5} - \frac{1}{19} a^{4} + \frac{7}{19} a^{3} + \frac{3}{19} a^{2} - \frac{5}{19} a + \frac{9}{19}$, $\frac{1}{247} a^{17} - \frac{5}{247} a^{16} - \frac{59}{247} a^{15} - \frac{42}{247} a^{14} + \frac{34}{247} a^{13} + \frac{81}{247} a^{12} + \frac{1}{247} a^{11} - \frac{98}{247} a^{10} - \frac{16}{247} a^{9} - \frac{31}{247} a^{8} - \frac{29}{247} a^{7} + \frac{20}{247} a^{6} - \frac{81}{247} a^{5} - \frac{53}{247} a^{4} - \frac{90}{247} a^{3} - \frac{15}{247} a^{2} + \frac{51}{247} a + \frac{103}{247}$, $\frac{1}{1729} a^{18} + \frac{20}{1729} a^{16} + \frac{807}{1729} a^{15} + \frac{136}{1729} a^{14} - \frac{607}{1729} a^{13} - \frac{647}{1729} a^{12} - \frac{860}{1729} a^{11} - \frac{571}{1729} a^{10} + \frac{27}{91} a^{9} - \frac{509}{1729} a^{8} + \frac{96}{1729} a^{7} + \frac{786}{1729} a^{6} + \frac{309}{1729} a^{5} - \frac{212}{1729} a^{4} + \frac{757}{1729} a^{3} - \frac{100}{247} a^{2} + \frac{332}{1729} a + \frac{463}{1729}$, $\frac{1}{1729} a^{19} - \frac{1}{1729} a^{17} + \frac{2}{1729} a^{16} + \frac{10}{1729} a^{15} - \frac{726}{1729} a^{14} + \frac{5}{91} a^{13} + \frac{13}{133} a^{12} + \frac{500}{1729} a^{11} + \frac{6}{91} a^{10} - \frac{446}{1729} a^{9} + \frac{565}{1729} a^{8} + \frac{758}{1729} a^{7} + \frac{526}{1729} a^{6} + \frac{397}{1729} a^{5} - \frac{678}{1729} a^{4} + \frac{1}{247} a^{3} - \frac{354}{1729} a^{2} + \frac{484}{1729} a + \frac{3}{247}$, $\frac{1}{12103} a^{20} + \frac{2}{12103} a^{19} - \frac{3}{12103} a^{18} + \frac{1}{1729} a^{17} + \frac{303}{12103} a^{16} + \frac{3000}{12103} a^{15} + \frac{4356}{12103} a^{14} + \frac{3995}{12103} a^{13} - \frac{1851}{12103} a^{12} + \frac{2750}{12103} a^{11} - \frac{863}{1729} a^{10} - \frac{2739}{12103} a^{9} + \frac{5874}{12103} a^{8} + \frac{3285}{12103} a^{7} + \frac{1837}{12103} a^{6} + \frac{751}{12103} a^{5} + \frac{69}{12103} a^{4} + \frac{64}{12103} a^{3} - \frac{97}{247} a^{2} - \frac{2867}{12103} a - \frac{2074}{12103}$, $\frac{1}{46633477757166732178466141473973809159219538227} a^{21} - \frac{222108992245185768539233796530930667319546}{6661925393880961739780877353424829879888505461} a^{20} - \frac{1729419589430360337418037337068395656404637}{6661925393880961739780877353424829879888505461} a^{19} + \frac{383657478673866036106982278871146604728052}{3587190596705133244497395497997985319939964479} a^{18} + \frac{34283204770408040253223568168287342323089680}{46633477757166732178466141473973809159219538227} a^{17} - \frac{5396863076377456414754794132405012815369946}{6661925393880961739780877353424829879888505461} a^{16} + \frac{15701340069958253725675768107198141520616944227}{46633477757166732178466141473973809159219538227} a^{15} + \frac{8326491895217010192409123588152334932978529610}{46633477757166732178466141473973809159219538227} a^{14} - \frac{6777855073020105915208705566610281699770333109}{46633477757166732178466141473973809159219538227} a^{13} + \frac{14698866739220473611883889657603746538700445030}{46633477757166732178466141473973809159219538227} a^{12} - \frac{17013808011548619126792684353298809915354978686}{46633477757166732178466141473973809159219538227} a^{11} - \frac{10787754453440428812991670721831484544272951115}{46633477757166732178466141473973809159219538227} a^{10} - \frac{1749234008033168676645806154159969624889901183}{46633477757166732178466141473973809159219538227} a^{9} + \frac{1265989287549897662296953166790402516036264221}{6661925393880961739780877353424829879888505461} a^{8} - \frac{17338297483220019451503539426168603841878054682}{46633477757166732178466141473973809159219538227} a^{7} + \frac{6216410367430633857044635583319699721452418189}{46633477757166732178466141473973809159219538227} a^{6} + \frac{13103282031554582137842560950156107373117170383}{46633477757166732178466141473973809159219538227} a^{5} - \frac{1524452840171120040282850091183990578435679096}{46633477757166732178466141473973809159219538227} a^{4} + \frac{11236321763740222083409684802742537115671698928}{46633477757166732178466141473973809159219538227} a^{3} - \frac{6379417569691829892613133126225694651665539}{879876938814466644876719650452336021872066759} a^{2} + \frac{21184273916244984289331456280418939584433200461}{46633477757166732178466141473973809159219538227} a - \frac{11811573347102022370527933879871355132818526217}{46633477757166732178466141473973809159219538227}$

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:		\( 8142471608830 \) (assuming GRH)

Galois group

$D_{11}$ (as 22T2):

A solvable group of order 22

The 7 conjugacy class representatives for $D_{11}$

Character table for $D_{11}$

Intermediate fields

\(\Q(\sqrt{4009}) \), 11.11.1035571956771279049.1 x11

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

Sibling fields

Degree 11 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	${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/7.2.0.1}{2} }^{11}$	${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/13.2.0.1}{2} }^{11}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{11}$	R	${\href{/LocalNumberField/23.2.0.1}{2} }^{11}$	${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$	${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{11}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{11}$

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