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Properties

Label 18.0.62065127981...9375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{44}\cdot 5^{9}\cdot 19^{9}$
Root discriminant $142.94$
Ramified primes $3, 5, 19$
Class number $30788064$ (GRH)
Class group $[2, 2, 7697016]$ (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![5291384827899, -1578496430709, 1795735413621, -452550480084, 272500787439, -58323184659, 24309351321, -4409563914, 1405734741, -213823058, 54623250, -6809130, 1424040, -139104, 23940, -1668, 234, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 234*x^16 - 1668*x^15 + 23940*x^14 - 139104*x^13 + 1424040*x^12 - 6809130*x^11 + 54623250*x^10 - 213823058*x^9 + 1405734741*x^8 - 4409563914*x^7 + 24309351321*x^6 - 58323184659*x^5 + 272500787439*x^4 - 452550480084*x^3 + 1795735413621*x^2 - 1578496430709*x + 5291384827899)
 
gp: K = bnfinit(x^18 - 9*x^17 + 234*x^16 - 1668*x^15 + 23940*x^14 - 139104*x^13 + 1424040*x^12 - 6809130*x^11 + 54623250*x^10 - 213823058*x^9 + 1405734741*x^8 - 4409563914*x^7 + 24309351321*x^6 - 58323184659*x^5 + 272500787439*x^4 - 452550480084*x^3 + 1795735413621*x^2 - 1578496430709*x + 5291384827899, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 234 x^{16} - 1668 x^{15} + 23940 x^{14} - 139104 x^{13} + 1424040 x^{12} - 6809130 x^{11} + 54623250 x^{10} - 213823058 x^{9} + 1405734741 x^{8} - 4409563914 x^{7} + 24309351321 x^{6} - 58323184659 x^{5} + 272500787439 x^{4} - 452550480084 x^{3} + 1795735413621 x^{2} - 1578496430709 x + 5291384827899 \)

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:  \(-620651279815192016958464288029880859375=-\,3^{44}\cdot 5^{9}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $142.94$
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:  $3, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2565=3^{3}\cdot 5\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{2565}(1,·)$, $\chi_{2565}(94,·)$, $\chi_{2565}(286,·)$, $\chi_{2565}(379,·)$, $\chi_{2565}(571,·)$, $\chi_{2565}(664,·)$, $\chi_{2565}(856,·)$, $\chi_{2565}(949,·)$, $\chi_{2565}(1141,·)$, $\chi_{2565}(1234,·)$, $\chi_{2565}(1426,·)$, $\chi_{2565}(1519,·)$, $\chi_{2565}(1711,·)$, $\chi_{2565}(1804,·)$, $\chi_{2565}(1996,·)$, $\chi_{2565}(2089,·)$, $\chi_{2565}(2281,·)$, $\chi_{2565}(2374,·)$$\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}{805475591210723794640684998072729366912252982995046127129153313527835009} a^{17} + \frac{381406682077455679292334258360897168563584341020961847053289123162193082}{805475591210723794640684998072729366912252982995046127129153313527835009} a^{16} + \frac{310173888426439330788437193835767899406894137174884323827539842272545582}{805475591210723794640684998072729366912252982995046127129153313527835009} a^{15} + \frac{316534757577341759569278269907274449816522002449810028504447520398144615}{805475591210723794640684998072729366912252982995046127129153313527835009} a^{14} + \frac{357256031080575943934019822928414961620441913105112516273937761400961217}{805475591210723794640684998072729366912252982995046127129153313527835009} a^{13} + \frac{158548718500711096027164133771936930278452481554045627009085054423781311}{805475591210723794640684998072729366912252982995046127129153313527835009} a^{12} + \frac{295489877763713027459910488917443785309452007792009607342974091816682085}{805475591210723794640684998072729366912252982995046127129153313527835009} a^{11} + \frac{223810171247144229818690479245722043568014678511247007285685768865029001}{805475591210723794640684998072729366912252982995046127129153313527835009} a^{10} - \frac{106711259107639377977227254734214664955118676673438655213018448556985991}{805475591210723794640684998072729366912252982995046127129153313527835009} a^{9} - \frac{353584008743677788337963809204284028979354391368944075524018182192841230}{805475591210723794640684998072729366912252982995046127129153313527835009} a^{8} + \frac{11216620482742141102163084128853373465296962601197372695735979472119260}{805475591210723794640684998072729366912252982995046127129153313527835009} a^{7} + \frac{334126772868174274337887570290334430335125605622654852223440502989908763}{805475591210723794640684998072729366912252982995046127129153313527835009} a^{6} - \frac{348792758059342382509679427931380818849518098694596575087275276195372531}{805475591210723794640684998072729366912252982995046127129153313527835009} a^{5} - \frac{193824707257663144693883133910545178551396643521075600082572920593758272}{805475591210723794640684998072729366912252982995046127129153313527835009} a^{4} + \frac{23011579703928156694178420602891875729717799910237496324646473921762700}{805475591210723794640684998072729366912252982995046127129153313527835009} a^{3} + \frac{375650421051066413351530022537299901704739809541182736901448552555939578}{805475591210723794640684998072729366912252982995046127129153313527835009} a^{2} - \frac{296739374670803291741352294687996228628224731586285921953575766075005710}{805475591210723794640684998072729366912252982995046127129153313527835009} a + \frac{4658807508971788064641190780837676404075456012933114044598808785958308}{15197652664353279144163867888164705036080244962170681643946288934487453}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{7697016}$, which has order $30788064$ (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:  \( 40934.03294431194 \) (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

\(\Q(\sqrt{-95}) \), \(\Q(\zeta_{9})^+\), 6.0.5625237375.5, \(\Q(\zeta_{27})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R R $18$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R $18$ $18$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ $18$ $18$ $18$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ $18$

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
$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]$
5Data not computed
$19$19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$