Global number field 22.22.78048218870425324004237696277333187889.1

• Label: 22.22.7804821887...7889.1
• Degree: $22$
• Signature: $[22, 0]$
• Discriminant: $7^{11}\cdot 23^{21}$
• Root discriminant: $52.77$
• Ramified primes: $7, 23$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{22}$ (as 22T1)

Normalized defining polynomial

\( x^{22} - x^{21} - 45 x^{20} + 45 x^{19} + 875 x^{18} - 875 x^{17} - 9613 x^{16} + 9613 x^{15} + 65459 x^{14} - 65459 x^{13} - 284877 x^{12} + 284877 x^{11} + 786739 x^{10} - 786739 x^{9} - 1318221 x^{8} + 1318221 x^{7} + 1207731 x^{6} - 1207731 x^{5} - 476237 x^{4} + 476237 x^{3} + 41907 x^{2} - 41907 x - 5197 \)

Invariants

Degree:		$22$
Signature:		$[22, 0]$
Discriminant:		\(78048218870425324004237696277333187889=7^{11}\cdot 23^{21}\)
Root discriminant:		$52.77$
Ramified primes:		$7, 23$
This field is Galois and abelian over $\Q$.
Conductor:		\(161=7\cdot 23\)
Dirichlet character group:		$\lbrace$$\chi_{161}(64,·)$, $\chi_{161}(1,·)$, $\chi_{161}(132,·)$, $\chi_{161}(71,·)$, $\chi_{161}(8,·)$, $\chi_{161}(76,·)$, $\chi_{161}(141,·)$, $\chi_{161}(78,·)$, $\chi_{161}(83,·)$, $\chi_{161}(20,·)$, $\chi_{161}(85,·)$, $\chi_{161}(153,·)$, $\chi_{161}(90,·)$, $\chi_{161}(29,·)$, $\chi_{161}(160,·)$, $\chi_{161}(97,·)$, $\chi_{161}(34,·)$, $\chi_{161}(36,·)$, $\chi_{161}(111,·)$, $\chi_{161}(50,·)$, $\chi_{161}(125,·)$, $\chi_{161}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{967} a^{12} - \frac{44}{967} a^{11} - \frac{24}{967} a^{10} + \frac{1}{967} a^{9} + \frac{216}{967} a^{8} - \frac{8}{967} a^{7} + \frac{71}{967} a^{6} + \frac{28}{967} a^{5} - \frac{254}{967} a^{4} - \frac{40}{967} a^{3} - \frac{185}{967} a^{2} + \frac{16}{967} a + \frac{128}{967}$, $\frac{1}{967} a^{13} - \frac{26}{967} a^{11} - \frac{88}{967} a^{10} + \frac{260}{967} a^{9} - \frac{174}{967} a^{8} - \frac{281}{967} a^{7} + \frac{251}{967} a^{6} + \frac{11}{967} a^{5} + \frac{388}{967} a^{4} - \frac{11}{967} a^{3} - \frac{388}{967} a^{2} - \frac{135}{967} a - \frac{170}{967}$, $\frac{1}{967} a^{14} - \frac{265}{967} a^{11} - \frac{364}{967} a^{10} - \frac{148}{967} a^{9} - \frac{467}{967} a^{8} + \frac{43}{967} a^{7} - \frac{77}{967} a^{6} + \frac{149}{967} a^{5} + \frac{154}{967} a^{4} - \frac{461}{967} a^{3} - \frac{110}{967} a^{2} + \frac{246}{967} a + \frac{427}{967}$, $\frac{1}{967} a^{15} - \frac{420}{967} a^{11} + \frac{261}{967} a^{10} - \frac{202}{967} a^{9} + \frac{230}{967} a^{8} - \frac{263}{967} a^{7} - \frac{376}{967} a^{6} - \frac{162}{967} a^{5} - \frac{81}{967} a^{4} - \frac{73}{967} a^{3} - \frac{429}{967} a^{2} - \frac{168}{967} a + \frac{75}{967}$, $\frac{1}{967} a^{16} + \frac{154}{967} a^{11} + \frac{355}{967} a^{10} - \frac{317}{967} a^{9} - \frac{441}{967} a^{8} + \frac{132}{967} a^{7} - \frac{319}{967} a^{6} + \frac{75}{967} a^{5} - \frac{383}{967} a^{4} + \frac{177}{967} a^{3} + \frac{459}{967} a^{2} + \frac{26}{967} a - \frac{392}{967}$, $\frac{1}{967} a^{17} + \frac{362}{967} a^{11} + \frac{478}{967} a^{10} + \frac{372}{967} a^{9} - \frac{254}{967} a^{8} - \frac{54}{967} a^{7} - \frac{222}{967} a^{6} + \frac{140}{967} a^{5} - \frac{354}{967} a^{4} - \frac{150}{967} a^{3} + \frac{473}{967} a^{2} + \frac{45}{967} a - \frac{372}{967}$, $\frac{1}{967} a^{18} - \frac{33}{967} a^{11} + \frac{357}{967} a^{10} + \frac{351}{967} a^{9} + \frac{81}{967} a^{8} - \frac{227}{967} a^{7} - \frac{420}{967} a^{6} + \frac{147}{967} a^{5} - \frac{67}{967} a^{4} + \frac{448}{967} a^{3} + \frac{292}{967} a^{2} - \frac{362}{967} a + \frac{80}{967}$, $\frac{1}{967} a^{19} - \frac{128}{967} a^{11} - \frac{441}{967} a^{10} + \frac{114}{967} a^{9} + \frac{132}{967} a^{8} + \frac{283}{967} a^{7} - \frac{411}{967} a^{6} - \frac{110}{967} a^{5} - \frac{198}{967} a^{4} - \frac{61}{967} a^{3} + \frac{302}{967} a^{2} - \frac{359}{967} a + \frac{356}{967}$, $\frac{1}{967} a^{20} - \frac{271}{967} a^{11} - \frac{57}{967} a^{10} + \frac{260}{967} a^{9} - \frac{112}{967} a^{8} - \frac{468}{967} a^{7} + \frac{275}{967} a^{6} - \frac{482}{967} a^{5} + \frac{305}{967} a^{4} + \frac{17}{967} a^{3} + \frac{136}{967} a^{2} + \frac{470}{967} a - \frac{55}{967}$, $\frac{1}{967} a^{21} - \frac{377}{967} a^{11} - \frac{442}{967} a^{10} + \frac{159}{967} a^{9} + \frac{48}{967} a^{8} + \frac{41}{967} a^{7} + \frac{386}{967} a^{6} + \frac{157}{967} a^{5} - \frac{160}{967} a^{4} - \frac{67}{967} a^{3} - \frac{348}{967} a^{2} + \frac{413}{967} a - \frac{124}{967}$

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:		\( 482091917601 \) (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

\(\Q(\sqrt{161}) \), \(\Q(\zeta_{23})^+\)

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	${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$	$22$	${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$	R	$22$	$22$	${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$	R	${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$	$22$	$22$	$22$	$22$	${\href{/LocalNumberField/47.2.0.1}{2} }^{11}$	$22$	$22$

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
7	Data not computed
23	Data not computed