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Properties

Label 18.2.17617410048...3344.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{16}\cdot 3^{32}\cdot 29^{9}$
Root discriminant $70.31$
Ramified primes $2, 3, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times D_9:C_3$ (as 18T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-42840647, -49246227, 20465379, 47849154, -1365048, -21336534, -1448754, 5669820, 482751, -978477, -68427, 114552, 3918, -8946, 126, 420, -27, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 27*x^16 + 420*x^15 + 126*x^14 - 8946*x^13 + 3918*x^12 + 114552*x^11 - 68427*x^10 - 978477*x^9 + 482751*x^8 + 5669820*x^7 - 1448754*x^6 - 21336534*x^5 - 1365048*x^4 + 47849154*x^3 + 20465379*x^2 - 49246227*x - 42840647)
 
gp: K = bnfinit(x^18 - 9*x^17 - 27*x^16 + 420*x^15 + 126*x^14 - 8946*x^13 + 3918*x^12 + 114552*x^11 - 68427*x^10 - 978477*x^9 + 482751*x^8 + 5669820*x^7 - 1448754*x^6 - 21336534*x^5 - 1365048*x^4 + 47849154*x^3 + 20465379*x^2 - 49246227*x - 42840647, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} - 27 x^{16} + 420 x^{15} + 126 x^{14} - 8946 x^{13} + 3918 x^{12} + 114552 x^{11} - 68427 x^{10} - 978477 x^{9} + 482751 x^{8} + 5669820 x^{7} - 1448754 x^{6} - 21336534 x^{5} - 1365048 x^{4} + 47849154 x^{3} + 20465379 x^{2} - 49246227 x - 42840647 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1761741004859375588383275406393344=2^{16}\cdot 3^{32}\cdot 29^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.31$
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:  $2, 3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{8} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{9} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{133444210029850886196597451983092159038786143356302287} a^{17} + \frac{17727596243463179854966757512953809960165195669471467}{133444210029850886196597451983092159038786143356302287} a^{16} - \frac{1665215870568949219793544714933874006289659397774504}{133444210029850886196597451983092159038786143356302287} a^{15} - \frac{3785891110701875991638500435832263099558354337946690}{44481403343283628732199150661030719679595381118767429} a^{14} - \frac{16228266003754041214171132885732842308765043289267792}{133444210029850886196597451983092159038786143356302287} a^{13} - \frac{13455784053076063643975387554888624022176223768898938}{133444210029850886196597451983092159038786143356302287} a^{12} - \frac{17878582229922915887416431960697792676538420023159365}{133444210029850886196597451983092159038786143356302287} a^{11} - \frac{185726636416767468777981422033106810551773598272109}{44481403343283628732199150661030719679595381118767429} a^{10} + \frac{1650755972406604556633955012899248376792064312161950}{12131291820895535108781586543917469003526013032391117} a^{9} + \frac{16187971901537736816630691103267527539848829707515394}{133444210029850886196597451983092159038786143356302287} a^{8} - \frac{20290287368654313733100501858144374964105242598008505}{133444210029850886196597451983092159038786143356302287} a^{7} + \frac{736136630905893175499606440831904529357551877991189}{2616553137840213454835244156531218804682081242280437} a^{6} + \frac{43773223231268194260400305839279009966362050076160527}{133444210029850886196597451983092159038786143356302287} a^{5} - \frac{23173543502131510911158865372782346447346456198990471}{133444210029850886196597451983092159038786143356302287} a^{4} + \frac{3110657221569348619500477070113598140317775082117622}{7849659413520640364505732469593656414046243726841311} a^{3} - \frac{9167877363038350809176901212312122227284761914191017}{133444210029850886196597451983092159038786143356302287} a^{2} - \frac{49460069072553848298210075765022647520455727810673815}{133444210029850886196597451983092159038786143356302287} a - \frac{29379705851499406330596818611814052118089543874968241}{133444210029850886196597451983092159038786143356302287}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 1763951815.7009294 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_9:C_3$ (as 18T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times D_9:C_3$
Character table for $C_2\times D_9:C_3$

Intermediate fields

\(\Q(\sqrt{29}) \), 3.1.108.1, 6.2.284473296.1, 9.1.11019960576.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 sibling: data not computed

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ $18$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.12.6.1$x^{12} + 146334 x^{6} - 20511149 x^{2} + 5353409889$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$