Global number field 18.0.6316806100079189177484411392000000000.1

• Label: 18.0.63168061000...0000.1
• Degree: $18$
• Signature: $[0, 9]$
• Discriminant: $-\,2^{18}\cdot 5^{9}\cdot 37^{16}$
• Root discriminant: $110.78$
• Ramified primes: $2, 5, 37$
• Class number: $146978$ (GRH)
• Class group: $[146978]$ (GRH)
• Galois group: $C_{18}$ (as 18T1)

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 14 x^{16} - 26 x^{15} + 346 x^{14} - 498 x^{13} + 5677 x^{12} - 6174 x^{11} + 71998 x^{10} - 56440 x^{9} + 707615 x^{8} - 471728 x^{7} + 5120273 x^{6} - 2764682 x^{5} + 26177811 x^{4} - 8570496 x^{3} + 82523855 x^{2} - 10820664 x + 116015941 \)

Invariants

Degree:		$18$
Signature:		$[0, 9]$
Discriminant:		\(-6316806100079189177484411392000000000=-\,2^{18}\cdot 5^{9}\cdot 37^{16}\)
Root discriminant:		$110.78$
Ramified primes:		$2, 5, 37$
This field is Galois and abelian over $\Q$.
Conductor:		\(740=2^{2}\cdot 5\cdot 37\)
Dirichlet character group:		$\lbrace$$\chi_{740}(641,·)$, $\chi_{740}(581,·)$, $\chi_{740}(441,·)$, $\chi_{740}(1,·)$, $\chi_{740}(201,·)$, $\chi_{740}(719,·)$, $\chi_{740}(81,·)$, $\chi_{740}(599,·)$, $\chi_{740}(601,·)$, $\chi_{740}(699,·)$, $\chi_{740}(219,·)$, $\chi_{740}(419,·)$, $\chi_{740}(359,·)$, $\chi_{740}(519,·)$, $\chi_{740}(181,·)$, $\chi_{740}(121,·)$, $\chi_{740}(379,·)$, $\chi_{740}(639,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{43} a^{15} - \frac{12}{43} a^{14} - \frac{7}{43} a^{13} - \frac{13}{43} a^{12} + \frac{5}{43} a^{11} - \frac{17}{43} a^{10} + \frac{15}{43} a^{9} + \frac{13}{43} a^{8} - \frac{16}{43} a^{7} + \frac{18}{43} a^{6} - \frac{14}{43} a^{5} - \frac{17}{43} a^{4} - \frac{7}{43} a^{3} - \frac{5}{43} a^{2} - \frac{5}{43} a + \frac{18}{43}$, $\frac{1}{1333} a^{16} - \frac{9}{1333} a^{15} - \frac{6}{31} a^{14} + \frac{52}{1333} a^{13} - \frac{636}{1333} a^{12} - \frac{260}{1333} a^{11} - \frac{380}{1333} a^{10} - \frac{114}{1333} a^{9} - \frac{536}{1333} a^{8} - \frac{331}{1333} a^{7} + \frac{40}{1333} a^{6} + \frac{27}{1333} a^{5} - \frac{101}{1333} a^{4} - \frac{628}{1333} a^{3} + \frac{582}{1333} a^{2} + \frac{261}{1333} a - \frac{634}{1333}$, $\frac{1}{105967220779219475239421609200903858202199827507144519} a^{17} - \frac{11113998480459540906003607431854720738939481801197}{105967220779219475239421609200903858202199827507144519} a^{16} + \frac{824832474413476216418107735358186710236691277123582}{105967220779219475239421609200903858202199827507144519} a^{15} + \frac{43749452225517368339118114585724778459757676538152910}{105967220779219475239421609200903858202199827507144519} a^{14} + \frac{18089109880166997155688900220866395330228841028092019}{105967220779219475239421609200903858202199827507144519} a^{13} - \frac{52945765546046931690638372132406044544958449132645110}{105967220779219475239421609200903858202199827507144519} a^{12} - \frac{9910528170906939589870313166145537954560161986340574}{105967220779219475239421609200903858202199827507144519} a^{11} - \frac{7508321394786244403872027238204532036715716367606677}{105967220779219475239421609200903858202199827507144519} a^{10} - \frac{15678589880710059238814731172453450977321049875498307}{105967220779219475239421609200903858202199827507144519} a^{9} - \frac{25533598258656339313623521883469736531944643038278705}{105967220779219475239421609200903858202199827507144519} a^{8} - \frac{21871848842480002356988040060686737005490565658210974}{105967220779219475239421609200903858202199827507144519} a^{7} - \frac{23461408757662490800619384278622815584381759601147710}{105967220779219475239421609200903858202199827507144519} a^{6} + \frac{33072768238786388985533711609754174640343781585835635}{105967220779219475239421609200903858202199827507144519} a^{5} - \frac{32018148598860485448963177150539333305010954993472052}{105967220779219475239421609200903858202199827507144519} a^{4} - \frac{11321423840291689568592262479373402424063487808455311}{105967220779219475239421609200903858202199827507144519} a^{3} + \frac{37122151651138010381297231081071954147186493348074885}{105967220779219475239421609200903858202199827507144519} a^{2} - \frac{190589442885034978090913669394708657064767315415550}{105967220779219475239421609200903858202199827507144519} a - \frac{390605376500897753691628440806128042478008861328841}{2464353971609755238126083934904740888423251802491733}$

Class group and class number

$C_{146978}$, which has order $146978$ (assuming GRH)

Unit group

Rank:		$8$
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:		\( 409151.3102125697 \) (assuming GRH)

Galois group

$C_{18}$ (as 18T1):

A cyclic group of order 18

The 18 conjugacy class representatives for $C_{18}$

Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-5}) \), 3.3.1369.1, 6.0.14993288000.2, 9.9.3512479453921.1

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	R	${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$	R	${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$	$18$	$18$	$18$	${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$	R	${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/43.1.0.1}{1} }^{18}$	${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$	$18$	$18$

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
5	Data not computed
37	Data not computed