Global number field 18.0.5333686073828961010482481752887525376.1

• Label: 18.0.53336860738...5376.1
• Degree: $18$
• Signature: $[0, 9]$
• Discriminant: $-\,2^{27}\cdot 3^{44}\cdot 7^{9}$
• Root discriminant: $109.75$
• Ramified primes: $2, 3, 7$
• Class number: $1668276$ (GRH)
• Class group: $[9, 185364]$ (GRH)
• Galois group: $C_{18}$ (as 18T1)

Normalized defining polynomial

\( x^{18} + 108 x^{16} + 5679 x^{14} + 187740 x^{12} + 4265379 x^{10} - 2 x^{9} + 68738148 x^{8} + 1026 x^{7} + 783654186 x^{6} - 54738 x^{5} + 6087814200 x^{4} + 592092 x^{3} + 29249550873 x^{2} - 1092690 x + 66331746447 \)

Invariants

Degree:		$18$
Signature:		$[0, 9]$
Discriminant:		\(-5333686073828961010482481752887525376=-\,2^{27}\cdot 3^{44}\cdot 7^{9}\)
Root discriminant:		$109.75$
Ramified primes:		$2, 3, 7$
This field is Galois and abelian over $\Q$.
Conductor:		\(1512=2^{3}\cdot 3^{3}\cdot 7\)
Dirichlet character group:		$\lbrace$$\chi_{1512}(1,·)$, $\chi_{1512}(517,·)$, $\chi_{1512}(1345,·)$, $\chi_{1512}(841,·)$, $\chi_{1512}(13,·)$, $\chi_{1512}(1357,·)$, $\chi_{1512}(337,·)$, $\chi_{1512}(853,·)$, $\chi_{1512}(1177,·)$, $\chi_{1512}(349,·)$, $\chi_{1512}(673,·)$, $\chi_{1512}(1189,·)$, $\chi_{1512}(169,·)$, $\chi_{1512}(685,·)$, $\chi_{1512}(1009,·)$, $\chi_{1512}(181,·)$, $\chi_{1512}(505,·)$, $\chi_{1512}(1021,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{12637632894072579680009057407652478792421501270592479344920746249} a^{17} - \frac{1599482807296844446929146398414402829935878742234027724612180788}{12637632894072579680009057407652478792421501270592479344920746249} a^{16} + \frac{6206336403759903200124951178691489230322286716211076025093866580}{12637632894072579680009057407652478792421501270592479344920746249} a^{15} + \frac{3570772108763475247952364975103591423832687402748279831844371538}{12637632894072579680009057407652478792421501270592479344920746249} a^{14} - \frac{5454471015075769767550366265922412581659492624632668300390513186}{12637632894072579680009057407652478792421501270592479344920746249} a^{13} + \frac{4362512807981056984850977924933887336887607986239421019824056845}{12637632894072579680009057407652478792421501270592479344920746249} a^{12} - \frac{4428615140988867140330195532573122381474424245116728938643341270}{12637632894072579680009057407652478792421501270592479344920746249} a^{11} + \frac{4807475829097379767806185866016072670497973112287398460036587179}{12637632894072579680009057407652478792421501270592479344920746249} a^{10} - \frac{3545052502017073000376229585335736217031103532669520828719089784}{12637632894072579680009057407652478792421501270592479344920746249} a^{9} + \frac{5572661625978483409204062363629736262476987077773184160712818942}{12637632894072579680009057407652478792421501270592479344920746249} a^{8} - \frac{1958226639250458163595092335319625027548931987047888984698755386}{12637632894072579680009057407652478792421501270592479344920746249} a^{7} - \frac{267332029851504817738627307824662927068060321351533858558821272}{12637632894072579680009057407652478792421501270592479344920746249} a^{6} - \frac{3998133452543279354096266511551798065697654352809776159225745974}{12637632894072579680009057407652478792421501270592479344920746249} a^{5} - \frac{5507082098875105425611474866242452636186982977337579861814563193}{12637632894072579680009057407652478792421501270592479344920746249} a^{4} - \frac{2159822366774270426572268491225275146627525925613907996312621301}{12637632894072579680009057407652478792421501270592479344920746249} a^{3} - \frac{2733042539062095635350014762334575585555761001528557646585622791}{12637632894072579680009057407652478792421501270592479344920746249} a^{2} + \frac{1307993558158410520384070149938104517764599719926348077069107817}{12637632894072579680009057407652478792421501270592479344920746249} a + \frac{2974454120486593571578751154338000327055016008868038435787180669}{12637632894072579680009057407652478792421501270592479344920746249}$

Class group and class number

$C_{9}\times C_{185364}$, which has order $1668276$ (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:		\( 40934.03294431194 \) (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

\(\Q(\sqrt{-14}) \), \(\Q(\zeta_{9})^+\), 6.0.1152216576.2, \(\Q(\zeta_{27})^+\)

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	${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$	R	$18$	${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$	$18$	$18$	${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$	$18$	$18$	$18$	${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$	${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$3$	3.9.22.8	$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$	$9$	$1$	$22$	$C_9$	$[2, 3]$
	3.9.22.8	$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$	$9$	$1$	$22$	$C_9$	$[2, 3]$
7	Data not computed