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Properties

Label 18.0.37394775151...3623.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{15}\cdot 31^{12}$
Root discriminant $49.94$
Ramified primes $7, 31$
Class number $259$ (GRH)
Class group $[259]$ (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![262144, -327680, 442368, -561152, 715776, -909568, 1156288, -65248, 112400, -4120, 11044, -270, 1203, -157, 115, -13, 11, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 11*x^16 - 13*x^15 + 115*x^14 - 157*x^13 + 1203*x^12 - 270*x^11 + 11044*x^10 - 4120*x^9 + 112400*x^8 - 65248*x^7 + 1156288*x^6 - 909568*x^5 + 715776*x^4 - 561152*x^3 + 442368*x^2 - 327680*x + 262144)
 
gp: K = bnfinit(x^18 - x^17 + 11*x^16 - 13*x^15 + 115*x^14 - 157*x^13 + 1203*x^12 - 270*x^11 + 11044*x^10 - 4120*x^9 + 112400*x^8 - 65248*x^7 + 1156288*x^6 - 909568*x^5 + 715776*x^4 - 561152*x^3 + 442368*x^2 - 327680*x + 262144, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 11 x^{16} - 13 x^{15} + 115 x^{14} - 157 x^{13} + 1203 x^{12} - 270 x^{11} + 11044 x^{10} - 4120 x^{9} + 112400 x^{8} - 65248 x^{7} + 1156288 x^{6} - 909568 x^{5} + 715776 x^{4} - 561152 x^{3} + 442368 x^{2} - 327680 x + 262144 \)

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:  \(-3739477515129074045367351773623=-\,7^{15}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.94$
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:  $7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(217=7\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{217}(1,·)$, $\chi_{217}(67,·)$, $\chi_{217}(5,·)$, $\chi_{217}(32,·)$, $\chi_{217}(211,·)$, $\chi_{217}(149,·)$, $\chi_{217}(87,·)$, $\chi_{217}(25,·)$, $\chi_{217}(156,·)$, $\chi_{217}(94,·)$, $\chi_{217}(160,·)$, $\chi_{217}(36,·)$, $\chi_{217}(129,·)$, $\chi_{217}(180,·)$, $\chi_{217}(118,·)$, $\chi_{217}(187,·)$, $\chi_{217}(125,·)$, $\chi_{217}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{32} a^{9} - \frac{1}{32} a^{8} - \frac{1}{32} a^{7} + \frac{15}{32} a^{6} - \frac{1}{32} a^{5} - \frac{1}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{12} - \frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{1}{64} a^{9} - \frac{1}{64} a^{8} + \frac{15}{64} a^{7} + \frac{31}{64} a^{6} + \frac{15}{32} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4992849152} a^{13} - \frac{21118249}{4992849152} a^{12} + \frac{2312067}{4992849152} a^{11} + \frac{98558523}{4992849152} a^{10} - \frac{244383845}{4992849152} a^{9} + \frac{253323}{4992849152} a^{8} + \frac{840800987}{4992849152} a^{7} - \frac{454542771}{2496424576} a^{6} + \frac{429943997}{1248212288} a^{5} + \frac{143683155}{624106144} a^{4} - \frac{95133817}{312053072} a^{3} + \frac{56586113}{156026536} a^{2} - \frac{24927859}{78013268} a + \frac{142722}{19503317}$, $\frac{1}{19971396608} a^{14} - \frac{1}{19971396608} a^{13} - \frac{95901653}{19971396608} a^{12} - \frac{41318701}{19971396608} a^{11} + \frac{330514451}{19971396608} a^{10} + \frac{985509891}{19971396608} a^{9} + \frac{1989085011}{19971396608} a^{8} - \frac{1292082487}{9985698304} a^{7} - \frac{2395924751}{4992849152} a^{6} - \frac{263730291}{2496424576} a^{5} - \frac{140103187}{1248212288} a^{4} + \frac{96325201}{624106144} a^{3} - \frac{77051349}{312053072} a^{2} - \frac{8395523}{78013268} a - \frac{1004531}{19503317}$, $\frac{1}{79885586432} a^{15} - \frac{1}{79885586432} a^{14} - \frac{5}{79885586432} a^{13} - \frac{557521533}{79885586432} a^{12} + \frac{183472835}{79885586432} a^{11} - \frac{765839053}{79885586432} a^{10} - \frac{1859604861}{79885586432} a^{9} - \frac{3719799583}{39942793216} a^{8} - \frac{4320790467}{19971396608} a^{7} - \frac{2781516231}{9985698304} a^{6} + \frac{949697925}{4992849152} a^{5} + \frac{322606469}{2496424576} a^{4} + \frac{601624367}{1248212288} a^{3} + \frac{103594813}{312053072} a^{2} + \frac{3228973}{19503317} a + \frac{197612}{19503317}$, $\frac{1}{319542345728} a^{16} - \frac{1}{319542345728} a^{15} - \frac{5}{319542345728} a^{14} + \frac{3}{319542345728} a^{13} + \frac{1301877443}{319542345728} a^{12} - \frac{4946906317}{319542345728} a^{11} - \frac{2005715837}{319542345728} a^{10} - \frac{107065375}{159771172864} a^{9} + \frac{5116827069}{79885586432} a^{8} + \frac{1385399545}{39942793216} a^{7} - \frac{7979094083}{19971396608} a^{6} + \frac{4015572021}{9985698304} a^{5} + \frac{997354143}{4992849152} a^{4} - \frac{389619571}{1248212288} a^{3} - \frac{14610051}{78013268} a^{2} + \frac{16145153}{39006634} a - \frac{7004288}{19503317}$, $\frac{1}{1278169382912} a^{17} - \frac{1}{1278169382912} a^{16} - \frac{5}{1278169382912} a^{15} + \frac{3}{1278169382912} a^{14} - \frac{61}{1278169382912} a^{13} - \frac{5147881421}{1278169382912} a^{12} - \frac{11023886973}{1278169382912} a^{11} + \frac{18161031265}{639084691456} a^{10} - \frac{6993202691}{319542345728} a^{9} + \frac{4149691673}{159771172864} a^{8} - \frac{1396821299}{79885586432} a^{7} + \frac{4079437141}{39942793216} a^{6} + \frac{1038887727}{19971396608} a^{5} - \frac{2256530083}{4992849152} a^{4} + \frac{12394061}{156026536} a^{3} - \frac{24013201}{78013268} a^{2} + \frac{3616505}{39006634} a + \frac{9062757}{19503317}$

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

Class group and class number

$C_{259}$, which has order $259$ (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:  \( \frac{147855}{639084691456} a^{17} - \frac{1626405}{639084691456} a^{16} + \frac{1922115}{639084691456} a^{15} - \frac{17003325}{639084691456} a^{14} + \frac{23213235}{639084691456} a^{13} - \frac{177869565}{639084691456} a^{12} + \frac{273707027}{639084691456} a^{11} - \frac{408227655}{159771172864} a^{10} + \frac{76145325}{79885586432} a^{9} - \frac{1038681375}{39942793216} a^{8} + \frac{301476345}{19971396608} a^{7} - \frac{2671296285}{9985698304} a^{6} + \frac{525328815}{2496424576} a^{5} - \frac{13854742733}{4992849152} a^{4} + \frac{20256135}{156026536} a^{3} - \frac{3992085}{39006634} a^{2} + \frac{1478550}{19503317} a - \frac{1182840}{19503317} \) (order $14$)
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:  \( 2999047.75971 \) (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

\(\Q(\sqrt{-7}) \), 3.3.47089.1, 3.3.961.1, \(\Q(\zeta_{7})^+\), 3.3.47089.2, 6.0.15521617447.1, 6.0.316767703.1, \(\Q(\zeta_{7})\), 6.0.15521617447.2, 9.9.104413920565969.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
7Data not computed
31Data not computed