Global number field 18.0.322803575628362753619753567652073327.1

• Label: 18.0.32280357562...3327.1
• Degree: $18$
• Signature: $[0, 9]$
• Discriminant: $-\,19^{16}\cdot 47^{9}$
• Root discriminant: $93.91$
• Ramified primes: $19, 47$
• Class number: $638885$ (GRH)
• Class group: $[638885]$ (GRH)
• Galois group: $C_{18}$ (as 18T1)

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 112 x^{16} - 602 x^{15} + 5587 x^{14} - 24267 x^{13} + 166052 x^{12} - 592865 x^{11} + 3262588 x^{10} - 9555455 x^{9} + 44088281 x^{8} - 103781941 x^{7} + 410585125 x^{6} - 741082426 x^{5} + 2545728128 x^{4} - 3182071487 x^{3} + 9558403988 x^{2} - 6299768998 x + 16620397249 \)

Invariants

Degree:		$18$
Signature:		$[0, 9]$
Discriminant:		\(-322803575628362753619753567652073327=-\,19^{16}\cdot 47^{9}\)
Root discriminant:		$93.91$
Ramified primes:		$19, 47$
This field is Galois and abelian over $\Q$.
Conductor:		\(893=19\cdot 47\)
Dirichlet character group:		$\lbrace$$\chi_{893}(704,·)$, $\chi_{893}(1,·)$, $\chi_{893}(518,·)$, $\chi_{893}(328,·)$, $\chi_{893}(330,·)$, $\chi_{893}(140,·)$, $\chi_{893}(845,·)$, $\chi_{893}(142,·)$, $\chi_{893}(847,·)$, $\chi_{893}(657,·)$, $\chi_{893}(283,·)$, $\chi_{893}(93,·)$, $\chi_{893}(612,·)$, $\chi_{893}(422,·)$, $\chi_{893}(424,·)$, $\chi_{893}(234,·)$, $\chi_{893}(377,·)$, $\chi_{893}(187,·)$$\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}{30842318782880889138534410535178268647280406237534945274681811} a^{17} - \frac{469750728531593939920515600839328318930248141270424177690833}{30842318782880889138534410535178268647280406237534945274681811} a^{16} + \frac{6457285124165235285174031746295470027323375841206324635807613}{30842318782880889138534410535178268647280406237534945274681811} a^{15} + \frac{11381217484306730209370460861547182156299915912127476524968097}{30842318782880889138534410535178268647280406237534945274681811} a^{14} - \frac{11573967423652202299112676806138818211523678272466883201279239}{30842318782880889138534410535178268647280406237534945274681811} a^{13} - \frac{11115207560916024212302199113996684310299562668279426328973952}{30842318782880889138534410535178268647280406237534945274681811} a^{12} + \frac{12583503049649324852524319484759346533801049791813288363353479}{30842318782880889138534410535178268647280406237534945274681811} a^{11} + \frac{332839145298256279797986575379621488021225974019812306002056}{833576183321105111852281365815628882358929898311755277694103} a^{10} - \frac{6393490498796527543922784105764280043663576541632772798416438}{30842318782880889138534410535178268647280406237534945274681811} a^{9} - \frac{10957658441084305966973247333655086093653399732232732096564252}{30842318782880889138534410535178268647280406237534945274681811} a^{8} - \frac{4390605159900554610162989312252137812938474010809679121544660}{30842318782880889138534410535178268647280406237534945274681811} a^{7} - \frac{4962971937074626170338292323116149763458865605945778998560529}{30842318782880889138534410535178268647280406237534945274681811} a^{6} + \frac{5162944625102315474771005425867494558228223878702515965446972}{30842318782880889138534410535178268647280406237534945274681811} a^{5} + \frac{6396597964200463551146523706873813771290661782267455277504142}{30842318782880889138534410535178268647280406237534945274681811} a^{4} - \frac{7939082058396512298225252011809833815075136077858145159664135}{30842318782880889138534410535178268647280406237534945274681811} a^{3} + \frac{12771648913256077996116901089625320246363023816111851216075160}{30842318782880889138534410535178268647280406237534945274681811} a^{2} + \frac{13806405400263415973584454711840114301413723849902823306826247}{30842318782880889138534410535178268647280406237534945274681811} a - \frac{13249725876982095381098200343711073853188098462242353303846828}{30842318782880889138534410535178268647280406237534945274681811}$

Class group and class number

$C_{638885}$, which has order $638885$ (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:		\( 22305.8950792 \) (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

\(\Q(\sqrt{-47}) \), 3.3.361.1, 6.0.13530317183.1, \(\Q(\zeta_{19})^+\)

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	${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$	$18$	${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$	${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$	$18$	${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$	R	$18$	$18$	${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$	${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$	$18$	$18$	R	${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$	${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
19	Data not computed
47	Data not computed