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Properties

Label 18.0.42004352444...8672.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 7^{12}\cdot 19^{15}$
Root discriminant $442.32$
Ramified primes $2, 3, 7, 19$
Class number $1013286144$ (GRH)
Class group $[2, 2, 2, 2, 52, 1217892]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![185027482261, -218068968786, 167812083477, 133588777454, -60150628077, -58803238512, 37334377373, -9350526024, 4026740871, -515196144, 171878073, -14341668, 3899592, -219576, 50028, -1778, 345, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 345*x^16 - 1778*x^15 + 50028*x^14 - 219576*x^13 + 3899592*x^12 - 14341668*x^11 + 171878073*x^10 - 515196144*x^9 + 4026740871*x^8 - 9350526024*x^7 + 37334377373*x^6 - 58803238512*x^5 - 60150628077*x^4 + 133588777454*x^3 + 167812083477*x^2 - 218068968786*x + 185027482261)
 
gp: K = bnfinit(x^18 - 6*x^17 + 345*x^16 - 1778*x^15 + 50028*x^14 - 219576*x^13 + 3899592*x^12 - 14341668*x^11 + 171878073*x^10 - 515196144*x^9 + 4026740871*x^8 - 9350526024*x^7 + 37334377373*x^6 - 58803238512*x^5 - 60150628077*x^4 + 133588777454*x^3 + 167812083477*x^2 - 218068968786*x + 185027482261, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 345 x^{16} - 1778 x^{15} + 50028 x^{14} - 219576 x^{13} + 3899592 x^{12} - 14341668 x^{11} + 171878073 x^{10} - 515196144 x^{9} + 4026740871 x^{8} - 9350526024 x^{7} + 37334377373 x^{6} - 58803238512 x^{5} - 60150628077 x^{4} + 133588777454 x^{3} + 167812083477 x^{2} - 218068968786 x + 185027482261 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-420043524447982694779997090972698756380285468672=-\,2^{18}\cdot 3^{27}\cdot 7^{12}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $442.32$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4788=2^{2}\cdot 3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4788}(1,·)$, $\chi_{4788}(683,·)$, $\chi_{4788}(2053,·)$, $\chi_{4788}(961,·)$, $\chi_{4788}(407,·)$, $\chi_{4788}(277,·)$, $\chi_{4788}(1367,·)$, $\chi_{4788}(1369,·)$, $\chi_{4788}(2459,·)$, $\chi_{4788}(1247,·)$, $\chi_{4788}(3299,·)$, $\chi_{4788}(4225,·)$, $\chi_{4788}(2857,·)$, $\chi_{4788}(4103,·)$, $\chi_{4788}(1775,·)$, $\chi_{4788}(3697,·)$, $\chi_{4788}(2615,·)$, $\chi_{4788}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} - \frac{3}{10} a^{5} - \frac{1}{5} a^{4} - \frac{1}{10} a^{3} + \frac{2}{5} a^{2} - \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{10} a^{10} - \frac{1}{10} a^{8} + \frac{2}{5} a^{7} - \frac{3}{10} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{3}{10}$, $\frac{1}{10} a^{11} + \frac{1}{10} a^{8} - \frac{1}{2} a^{7} + \frac{2}{5} a^{6} - \frac{1}{2} a^{5} - \frac{1}{10} a^{4} + \frac{3}{10} a^{3} + \frac{1}{10} a^{2} + \frac{1}{10}$, $\frac{1}{370} a^{12} - \frac{2}{185} a^{11} - \frac{13}{370} a^{10} - \frac{1}{370} a^{9} - \frac{7}{37} a^{8} - \frac{62}{185} a^{7} + \frac{49}{185} a^{6} - \frac{67}{370} a^{5} + \frac{57}{370} a^{4} + \frac{133}{370} a^{3} - \frac{62}{185} a^{2} + \frac{151}{370} a - \frac{29}{74}$, $\frac{1}{370} a^{13} + \frac{4}{185} a^{11} - \frac{8}{185} a^{10} - \frac{71}{370} a^{8} + \frac{157}{370} a^{7} - \frac{4}{185} a^{6} - \frac{10}{37} a^{5} - \frac{83}{370} a^{4} - \frac{73}{370} a^{3} - \frac{49}{370} a^{2} - \frac{59}{370} a - \frac{173}{370}$, $\frac{1}{370} a^{14} + \frac{8}{185} a^{11} - \frac{7}{370} a^{10} + \frac{11}{370} a^{9} + \frac{51}{370} a^{8} + \frac{11}{185} a^{7} - \frac{181}{370} a^{6} + \frac{157}{370} a^{5} + \frac{10}{37} a^{4} - \frac{3}{370} a^{3} - \frac{14}{37} a^{2} + \frac{5}{74} a - \frac{12}{185}$, $\frac{1}{10730} a^{15} + \frac{2}{5365} a^{14} - \frac{11}{10730} a^{13} + \frac{7}{5365} a^{12} + \frac{347}{10730} a^{11} - \frac{1}{58} a^{10} + \frac{49}{2146} a^{9} - \frac{21}{290} a^{8} - \frac{9}{5365} a^{7} + \frac{309}{2146} a^{6} - \frac{721}{5365} a^{5} - \frac{354}{1073} a^{4} + \frac{607}{10730} a^{3} - \frac{688}{5365} a^{2} + \frac{1089}{10730} a - \frac{33}{74}$, $\frac{1}{26735414803020625450790882197460} a^{16} + \frac{163329529192236267076075311}{13367707401510312725395441098730} a^{15} - \frac{504993411834226067858083055}{2673541480302062545079088219746} a^{14} - \frac{217679179343679362681730704}{1336770740151031272539544109873} a^{13} + \frac{5501191175216720697892856191}{13367707401510312725395441098730} a^{12} - \frac{81751769088502096257826396}{36128938923000845203771462429} a^{11} - \frac{992502217149041942695269881}{2673541480302062545079088219746} a^{10} + \frac{316955546171444144281099053534}{6683853700755156362697720549365} a^{9} + \frac{4399418671715389770998599022419}{26735414803020625450790882197460} a^{8} - \frac{451801116553672567384206734504}{1336770740151031272539544109873} a^{7} - \frac{6682696024843181527348778041261}{13367707401510312725395441098730} a^{6} + \frac{1535363792348435082072784030326}{6683853700755156362697720549365} a^{5} - \frac{78146157595730476374207020625}{5347082960604125090158176439492} a^{4} - \frac{2319523314532400370607764786664}{6683853700755156362697720549365} a^{3} + \frac{1072183882038474241311723412773}{2673541480302062545079088219746} a^{2} - \frac{1233519494885703840881312149399}{2673541480302062545079088219746} a + \frac{119915150851543765521555932283}{921910855276573291406582144740}$, $\frac{1}{26034707112189941936654534241222707205558704260} a^{17} - \frac{261626197465273}{26034707112189941936654534241222707205558704260} a^{16} + \frac{266815432670048608821771951807814979943989}{6508676778047485484163633560305676801389676065} a^{15} - \frac{666385736808593112881049999033128855941753}{6508676778047485484163633560305676801389676065} a^{14} + \frac{1296217450873508022491123526975501775730599}{1301735355609497096832726712061135360277935213} a^{13} + \frac{7938650024794891421730785958328502435248062}{6508676778047485484163633560305676801389676065} a^{12} - \frac{193196084230965934223896681306583608505472691}{13017353556094970968327267120611353602779352130} a^{11} + \frac{80740247375830888221825793932397763670836395}{2603470711218994193665453424122270720555870426} a^{10} - \frac{827403083516412591998324539758737247676371391}{26034707112189941936654534241222707205558704260} a^{9} - \frac{3334499363122150455356686132681003780724478469}{26034707112189941936654534241222707205558704260} a^{8} + \frac{1689030064931834633534549986619301672540033678}{6508676778047485484163633560305676801389676065} a^{7} + \frac{534044180704611756353483282380363128109549839}{1301735355609497096832726712061135360277935213} a^{6} - \frac{1739865927602609288374406380063366649308902163}{5206941422437988387330906848244541441111740852} a^{5} + \frac{10185840633333484101062353018025657066297217483}{26034707112189941936654534241222707205558704260} a^{4} + \frac{2082936981798989957962397679248716929405992653}{13017353556094970968327267120611353602779352130} a^{3} - \frac{995136203967866421624179331313916807635861001}{2603470711218994193665453424122270720555870426} a^{2} - \frac{1755333443703873132663232157148990244089601323}{5206941422437988387330906848244541441111740852} a + \frac{2633543316806937678270193358952858279766763}{21896305392926780434528624256705388734700340}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{52}\times C_{1217892}$, which has order $1013286144$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 42198260.232521206 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-57}) \), 3.3.1432809.2, 3.3.1432809.1, 3.3.29241.2, \(\Q(\zeta_{7})^+\), 6.0.7489131067994688.4, 6.0.7489131067994688.3, 6.0.3119171623488.3, 6.0.28457497152.2, 9.9.2941473244627851129.4

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
7Data not computed
19Data not computed