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Properties

Label 18.0.11192845998...0351.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 7^{12}\cdot 13^{9}$
Root discriminant $68.56$
Ramified primes $3, 7, 13$
Class number $31600$ (GRH)
Class group $[2, 10, 1580]$ (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![34082749, -10143966, 31588815, -10599756, 17344149, -3216606, 4002937, -365916, 573705, -40258, 83169, -17568, 12940, -2976, 1206, -212, 57, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 57*x^16 - 212*x^15 + 1206*x^14 - 2976*x^13 + 12940*x^12 - 17568*x^11 + 83169*x^10 - 40258*x^9 + 573705*x^8 - 365916*x^7 + 4002937*x^6 - 3216606*x^5 + 17344149*x^4 - 10599756*x^3 + 31588815*x^2 - 10143966*x + 34082749)
 
gp: K = bnfinit(x^18 - 6*x^17 + 57*x^16 - 212*x^15 + 1206*x^14 - 2976*x^13 + 12940*x^12 - 17568*x^11 + 83169*x^10 - 40258*x^9 + 573705*x^8 - 365916*x^7 + 4002937*x^6 - 3216606*x^5 + 17344149*x^4 - 10599756*x^3 + 31588815*x^2 - 10143966*x + 34082749, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 57 x^{16} - 212 x^{15} + 1206 x^{14} - 2976 x^{13} + 12940 x^{12} - 17568 x^{11} + 83169 x^{10} - 40258 x^{9} + 573705 x^{8} - 365916 x^{7} + 4002937 x^{6} - 3216606 x^{5} + 17344149 x^{4} - 10599756 x^{3} + 31588815 x^{2} - 10143966 x + 34082749 \)

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:  \(-1119284599813901503934006266380351=-\,3^{27}\cdot 7^{12}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.56$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(819=3^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{819}(1,·)$, $\chi_{819}(389,·)$, $\chi_{819}(779,·)$, $\chi_{819}(781,·)$, $\chi_{819}(79,·)$, $\chi_{819}(274,·)$, $\chi_{819}(662,·)$, $\chi_{819}(155,·)$, $\chi_{819}(352,·)$, $\chi_{819}(547,·)$, $\chi_{819}(233,·)$, $\chi_{819}(235,·)$, $\chi_{819}(428,·)$, $\chi_{819}(625,·)$, $\chi_{819}(116,·)$, $\chi_{819}(506,·)$, $\chi_{819}(508,·)$, $\chi_{819}(701,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{8} a^{2} + \frac{1}{8} a - \frac{1}{8}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{16} a^{5} + \frac{3}{16} a^{4} + \frac{7}{16} a^{3} - \frac{5}{16} a + \frac{3}{16}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} + \frac{1}{16} a^{6} + \frac{3}{16} a^{5} - \frac{1}{16} a^{4} + \frac{3}{16} a^{2} - \frac{5}{16} a - \frac{1}{2}$, $\frac{1}{2032} a^{14} + \frac{5}{1016} a^{13} - \frac{3}{1016} a^{12} + \frac{81}{2032} a^{11} - \frac{47}{2032} a^{10} + \frac{53}{2032} a^{9} - \frac{99}{1016} a^{8} - \frac{69}{2032} a^{7} - \frac{295}{2032} a^{6} + \frac{1}{1016} a^{5} + \frac{49}{2032} a^{4} - \frac{319}{1016} a^{3} - \frac{773}{2032} a^{2} - \frac{495}{2032} a + \frac{213}{2032}$, $\frac{1}{2032} a^{15} + \frac{21}{2032} a^{13} + \frac{7}{1016} a^{12} + \frac{2}{127} a^{11} + \frac{15}{2032} a^{10} - \frac{93}{2032} a^{9} + \frac{133}{2032} a^{8} - \frac{113}{2032} a^{7} + \frac{285}{2032} a^{6} - \frac{225}{2032} a^{5} + \frac{99}{508} a^{4} - \frac{435}{1016} a^{3} - \frac{32}{127} a^{2} + \frac{591}{2032} a - \frac{987}{2032}$, $\frac{1}{735584} a^{16} + \frac{1}{22987} a^{15} + \frac{25}{367792} a^{14} + \frac{2403}{183896} a^{13} + \frac{2299}{183896} a^{12} - \frac{1735}{367792} a^{11} + \frac{17419}{367792} a^{10} + \frac{4377}{91948} a^{9} - \frac{88721}{735584} a^{8} + \frac{16765}{367792} a^{7} + \frac{20659}{183896} a^{6} - \frac{65857}{367792} a^{5} + \frac{125491}{735584} a^{4} - \frac{963}{2032} a^{3} - \frac{42781}{91948} a^{2} - \frac{11841}{91948} a + \frac{139693}{735584}$, $\frac{1}{1750157587789156692139265884310717885553561664} a^{17} + \frac{427882917777493552038724585350767751131}{1750157587789156692139265884310717885553561664} a^{16} - \frac{9658096943754520896461571190860076436969}{218769698473644586517408235538839735694195208} a^{15} - \frac{6045081356683188558159570582555009963528}{27346212309205573314676029442354966961774401} a^{14} - \frac{27167379857512928944580750821468909738643841}{875078793894578346069632942155358942776780832} a^{13} - \frac{18959878146850857254991676044037111339988121}{875078793894578346069632942155358942776780832} a^{12} + \frac{40790700928035443242954796804492943912190591}{875078793894578346069632942155358942776780832} a^{11} + \frac{14195615941626770124863432081842222616817529}{875078793894578346069632942155358942776780832} a^{10} + \frac{68482735114368007399162893749606559689485983}{1750157587789156692139265884310717885553561664} a^{9} - \frac{68196582861204317359906760018874051809185683}{1750157587789156692139265884310717885553561664} a^{8} + \frac{91601480307354788075001471624073292885089495}{875078793894578346069632942155358942776780832} a^{7} + \frac{200439135049555978639732348021107985008749607}{875078793894578346069632942155358942776780832} a^{6} - \frac{99678712610561748545909882995366170889862305}{1750157587789156692139265884310717885553561664} a^{5} - \frac{403702895139107038876084772880434227489153171}{1750157587789156692139265884310717885553561664} a^{4} - \frac{238076180263966163980609012179453887556626557}{875078793894578346069632942155358942776780832} a^{3} + \frac{88961212096785010116998709410118989396163131}{875078793894578346069632942155358942776780832} a^{2} - \frac{486267483189150555179431180671269032991730951}{1750157587789156692139265884310717885553561664} a + \frac{282833716097998129374244653468531626511140871}{1750157587789156692139265884310717885553561664}$

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

Class group and class number

$C_{2}\times C_{10}\times C_{1580}$, which has order $31600$ (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:  \( 54408.4888887 \) (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

\(\Q(\sqrt{-39}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{7})^+\), 6.0.43243551.1, 6.0.103827765951.7, 6.0.103827765951.8, 6.0.142424919.1, 9.9.62523502209.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
7Data not computed
$13$13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$