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Properties

Label 18.0.16373578698...5872.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{24}\cdot 7^{9}\cdot 19^{17}$
Root discriminant $369.36$
Ramified primes $2, 3, 7, 19$
Class number $428224512$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 4, 418188]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18942165498853, 9476448749994, 5274173007969, 994931059476, 337785531834, 17871018366, 21610704006, 2435565198, 1621667955, -68927858, 33094599, -1431156, 3097703, 3648, 44460, -76, 285, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 285*x^16 - 76*x^15 + 44460*x^14 + 3648*x^13 + 3097703*x^12 - 1431156*x^11 + 33094599*x^10 - 68927858*x^9 + 1621667955*x^8 + 2435565198*x^7 + 21610704006*x^6 + 17871018366*x^5 + 337785531834*x^4 + 994931059476*x^3 + 5274173007969*x^2 + 9476448749994*x + 18942165498853)
 
gp: K = bnfinit(x^18 + 285*x^16 - 76*x^15 + 44460*x^14 + 3648*x^13 + 3097703*x^12 - 1431156*x^11 + 33094599*x^10 - 68927858*x^9 + 1621667955*x^8 + 2435565198*x^7 + 21610704006*x^6 + 17871018366*x^5 + 337785531834*x^4 + 994931059476*x^3 + 5274173007969*x^2 + 9476448749994*x + 18942165498853, 1)
 

Normalized defining polynomial

\( x^{18} + 285 x^{16} - 76 x^{15} + 44460 x^{14} + 3648 x^{13} + 3097703 x^{12} - 1431156 x^{11} + 33094599 x^{10} - 68927858 x^{9} + 1621667955 x^{8} + 2435565198 x^{7} + 21610704006 x^{6} + 17871018366 x^{5} + 337785531834 x^{4} + 994931059476 x^{3} + 5274173007969 x^{2} + 9476448749994 x + 18942165498853 \)

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:  \(-16373578698382653365249859609237042549755215872=-\,2^{18}\cdot 3^{24}\cdot 7^{9}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $369.36$
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$$\chi_{4788}(1,·)$, $\chi_{4788}(2437,·)$, $\chi_{4788}(4423,·)$, $\chi_{4788}(4171,·)$, $\chi_{4788}(4621,·)$, $\chi_{4788}(1261,·)$, $\chi_{4788}(85,·)$, $\chi_{4788}(1849,·)$, $\chi_{4788}(4591,·)$, $\chi_{4788}(223,·)$, $\chi_{4788}(2407,·)$, $\chi_{4788}(1063,·)$, $\chi_{4788}(169,·)$, $\chi_{4788}(3499,·)$, $\chi_{4788}(3949,·)$, $\chi_{4788}(559,·)$, $\chi_{4788}(505,·)$, $\chi_{4788}(2491,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19} a^{9}$, $\frac{1}{133} a^{10} + \frac{2}{133} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} + \frac{2}{7} a^{2} - \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{133} a^{11} - \frac{1}{133} a^{9} + \frac{1}{7} a^{8} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} + \frac{3}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{133} a^{12} + \frac{1}{7} a^{6} + \frac{3}{7}$, $\frac{1}{931} a^{13} - \frac{3}{931} a^{10} + \frac{1}{931} a^{9} - \frac{20}{49} a^{8} + \frac{3}{7} a^{7} - \frac{17}{49} a^{6} + \frac{2}{49} a^{5} + \frac{4}{49} a^{4} + \frac{13}{49} a^{3} - \frac{6}{49} a^{2} + \frac{20}{49} a + \frac{5}{49}$, $\frac{1}{931} a^{14} - \frac{3}{931} a^{11} + \frac{1}{931} a^{10} + \frac{12}{931} a^{9} + \frac{3}{7} a^{8} - \frac{17}{49} a^{7} + \frac{2}{49} a^{6} + \frac{4}{49} a^{5} + \frac{13}{49} a^{4} - \frac{6}{49} a^{3} + \frac{20}{49} a^{2} + \frac{5}{49} a$, $\frac{1}{6517} a^{15} + \frac{3}{6517} a^{14} + \frac{2}{6517} a^{13} + \frac{4}{6517} a^{12} - \frac{1}{6517} a^{11} + \frac{23}{6517} a^{10} - \frac{81}{6517} a^{9} - \frac{8}{343} a^{8} + \frac{2}{7} a^{7} - \frac{136}{343} a^{6} - \frac{160}{343} a^{5} - \frac{22}{343} a^{4} + \frac{6}{49} a^{3} + \frac{95}{343} a^{2} + \frac{27}{343} a - \frac{116}{343}$, $\frac{1}{6517} a^{16} - \frac{2}{6517} a^{13} - \frac{13}{6517} a^{12} + \frac{5}{6517} a^{11} + \frac{4}{6517} a^{10} + \frac{18}{931} a^{9} - \frac{123}{343} a^{8} - \frac{157}{343} a^{7} + \frac{66}{343} a^{6} + \frac{45}{343} a^{5} + \frac{3}{343} a^{4} - \frac{24}{343} a^{3} - \frac{167}{343} a^{2} + \frac{34}{343} a + \frac{103}{343}$, $\frac{1}{1129634803908312070705988712403034964764839697549054032442484133647248750576883941408078375809} a^{17} + \frac{57220258019235075028759100139113798607593496230479500688938216950293686176432794286395073}{1129634803908312070705988712403034964764839697549054032442484133647248750576883941408078375809} a^{16} + \frac{1920579584073654976964099156069764787674088875709191740428212473468790562978892072144990}{1129634803908312070705988712403034964764839697549054032442484133647248750576883941408078375809} a^{15} + \frac{495848189176222581908204290336167008636058688159733971932459994885132279603048649677075108}{1129634803908312070705988712403034964764839697549054032442484133647248750576883941408078375809} a^{14} + \frac{502309075338179061186428802296424161501795327626735524211758742866767439801330597538865763}{1129634803908312070705988712403034964764839697549054032442484133647248750576883941408078375809} a^{13} - \frac{470755328496474600405058665255693953362507764789697121387787117988371092646376397361543157}{161376400558330295815141244629004994966405671078436290348926304806749821510983420201154053687} a^{12} - \frac{305218815004795607105588457460512083637594036623204843938866728934901997394602059768652453}{161376400558330295815141244629004994966405671078436290348926304806749821510983420201154053687} a^{11} + \frac{1232700613677598090380452067728660666702139721828552382737214551986367866260795719622422906}{1129634803908312070705988712403034964764839697549054032442484133647248750576883941408078375809} a^{10} - \frac{20732870145287810545562696314282892285778512235110347188990825151088383364280979160114839636}{1129634803908312070705988712403034964764839697549054032442484133647248750576883941408078375809} a^{9} - \frac{16399691051894547955284190678446117379080711922225763325755056038091996990527942714480737586}{59454463363595372142420458547528156040254720923634422760130743876170986872467575863583072411} a^{8} - \frac{14967061165509452846283151419048445517072941450281301981320258534767685111542901893654132531}{59454463363595372142420458547528156040254720923634422760130743876170986872467575863583072411} a^{7} - \frac{13704325657611147394168255396654820249104648240026770961977165677359654249728258757361940800}{59454463363595372142420458547528156040254720923634422760130743876170986872467575863583072411} a^{6} - \frac{4299593973695109067843508270491450974642307995048405204610900526635691240834581371699429}{86041191553683606573690967507276636816577020149977456961115403583460183607044248717196921} a^{5} - \frac{25428483382961075184789076961138388738567407911483442523697354361719729141356827679277377439}{59454463363595372142420458547528156040254720923634422760130743876170986872467575863583072411} a^{4} + \frac{20374832591199838687152144063681612363692128608364431631485910560588979331478579657538586072}{59454463363595372142420458547528156040254720923634422760130743876170986872467575863583072411} a^{3} - \frac{756309203441752699195631722151451480255585802807020693326046803674061462182696759294864322}{59454463363595372142420458547528156040254720923634422760130743876170986872467575863583072411} a^{2} + \frac{2805671204578148155788377501629809536708868572806477938326585001356576397957337900444922845}{8493494766227910306060065506789736577179245846233488965732963410881569553209653694797581773} a + \frac{7704170191470293148559554186698817369791323016880286765359245590876061537929296125770344374}{59454463363595372142420458547528156040254720923634422760130743876170986872467575863583072411}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{418188}$, which has order $428224512$ (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:  \( 22027035.20428972 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-133}) \), 3.3.361.1, 6.0.54355325248.1, 9.9.9025761726072081.2

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 $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ R $18$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ $18$

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
2Data not computed
3Data not computed
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
19Data not computed