Global number field 22.22.86534669543385676516186776267386878120889.1

• Label: 22.22.8653466954...0889.1
• Degree: $22$
• Signature: $[22, 0]$
• Discriminant: $89^{21}$
• Root discriminant: $72.57$
• Ramified prime: $89$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{22}$ (as 22T1)

Normalized defining polynomial

\( x^{22} - x^{21} - 42 x^{20} + 37 x^{19} + 703 x^{18} - 539 x^{17} - 6079 x^{16} + 3987 x^{15} + 29502 x^{14} - 16237 x^{13} - 81976 x^{12} + 36419 x^{11} + 128582 x^{10} - 41490 x^{9} - 109384 x^{8} + 20339 x^{7} + 45984 x^{6} - 3224 x^{5} - 7518 x^{4} + 419 x^{3} + 376 x^{2} - 39 x + 1 \)

Invariants

Degree:		$22$
Signature:		$[22, 0]$
Discriminant:		\(86534669543385676516186776267386878120889=89^{21}\)
Root discriminant:		$72.57$
Ramified primes:		$89$
This field is Galois and abelian over $\Q$.
Conductor:		\(89\)
Dirichlet character group:		$\lbrace$$\chi_{89}(64,·)$, $\chi_{89}(1,·)$, $\chi_{89}(2,·)$, $\chi_{89}(67,·)$, $\chi_{89}(4,·)$, $\chi_{89}(8,·)$, $\chi_{89}(73,·)$, $\chi_{89}(11,·)$, $\chi_{89}(78,·)$, $\chi_{89}(16,·)$, $\chi_{89}(81,·)$, $\chi_{89}(85,·)$, $\chi_{89}(22,·)$, $\chi_{89}(87,·)$, $\chi_{89}(88,·)$, $\chi_{89}(25,·)$, $\chi_{89}(32,·)$, $\chi_{89}(39,·)$, $\chi_{89}(44,·)$, $\chi_{89}(45,·)$, $\chi_{89}(50,·)$, $\chi_{89}(57,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{233} a^{20} + \frac{84}{233} a^{19} + \frac{14}{233} a^{18} + \frac{88}{233} a^{17} + \frac{110}{233} a^{16} + \frac{73}{233} a^{15} + \frac{38}{233} a^{14} - \frac{111}{233} a^{13} - \frac{48}{233} a^{12} - \frac{97}{233} a^{11} + \frac{35}{233} a^{10} + \frac{48}{233} a^{9} + \frac{57}{233} a^{8} + \frac{84}{233} a^{7} + \frac{44}{233} a^{6} + \frac{106}{233} a^{5} + \frac{64}{233} a^{4} - \frac{59}{233} a^{3} + \frac{91}{233} a^{2} - \frac{47}{233} a - \frac{57}{233}$, $\frac{1}{954607074366869591183484326023} a^{21} + \frac{974357976834387703073140093}{954607074366869591183484326023} a^{20} + \frac{250914879844143521688030178984}{954607074366869591183484326023} a^{19} - \frac{445267276841925482438210739268}{954607074366869591183484326023} a^{18} + \frac{336144782368651006628589173780}{954607074366869591183484326023} a^{17} - \frac{292539787509977731856864311185}{954607074366869591183484326023} a^{16} + \frac{4355636153649338046378276929}{954607074366869591183484326023} a^{15} + \frac{350529344029086625854032265041}{954607074366869591183484326023} a^{14} + \frac{408221696336649892588566566770}{954607074366869591183484326023} a^{13} + \frac{389881362606937238550825286449}{954607074366869591183484326023} a^{12} - \frac{174773215136737706165068012532}{954607074366869591183484326023} a^{11} - \frac{402753855959321734343945356350}{954607074366869591183484326023} a^{10} - \frac{66253240909124749845778927081}{954607074366869591183484326023} a^{9} + \frac{274378119840618050008227059445}{954607074366869591183484326023} a^{8} - \frac{433315956799765023643542034437}{954607074366869591183484326023} a^{7} + \frac{335986110149943143769734998007}{954607074366869591183484326023} a^{6} - \frac{2917762136055709378332632816}{954607074366869591183484326023} a^{5} - \frac{10610830125780766385728879144}{954607074366869591183484326023} a^{4} + \frac{230249563679837404550757924981}{954607074366869591183484326023} a^{3} - \frac{58788500813808918633350187270}{954607074366869591183484326023} a^{2} - \frac{301733620272632258365507155630}{954607074366869591183484326023} a + \frac{383718066492168893133978688348}{954607074366869591183484326023}$

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:		\( 15755205659500 \) (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

\(\Q(\sqrt{89}) \), 11.11.31181719929966183601.1

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	${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$	$22$	${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$	$22$	${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$	$22$	${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$	$22$	$22$	$22$	$22$	${\href{/LocalNumberField/37.2.0.1}{2} }^{11}$	$22$	$22$	${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$	$22$

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
89	Data not computed