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Properties

Label 18.18.1402061561...1489.1
Degree $18$
Signature $[18, 0]$
Discriminant $19^{6}\cdot 1129^{9}$
Root discriminant $89.66$
Ramified primes $19, 1129$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2^2:D_9$ (as 18T39)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![269671, -568486, -2029989, 2797609, 3540642, -5139518, -1433009, 3486165, -226188, -1014925, 226382, 128965, -41170, -6917, 2903, 151, -89, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 89*x^16 + 151*x^15 + 2903*x^14 - 6917*x^13 - 41170*x^12 + 128965*x^11 + 226382*x^10 - 1014925*x^9 - 226188*x^8 + 3486165*x^7 - 1433009*x^6 - 5139518*x^5 + 3540642*x^4 + 2797609*x^3 - 2029989*x^2 - 568486*x + 269671)
 
gp: K = bnfinit(x^18 - x^17 - 89*x^16 + 151*x^15 + 2903*x^14 - 6917*x^13 - 41170*x^12 + 128965*x^11 + 226382*x^10 - 1014925*x^9 - 226188*x^8 + 3486165*x^7 - 1433009*x^6 - 5139518*x^5 + 3540642*x^4 + 2797609*x^3 - 2029989*x^2 - 568486*x + 269671, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 89 x^{16} + 151 x^{15} + 2903 x^{14} - 6917 x^{13} - 41170 x^{12} + 128965 x^{11} + 226382 x^{10} - 1014925 x^{9} - 226188 x^{8} + 3486165 x^{7} - 1433009 x^{6} - 5139518 x^{5} + 3540642 x^{4} + 2797609 x^{3} - 2029989 x^{2} - 568486 x + 269671 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(140206156168767266840209098851131489=19^{6}\cdot 1129^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $89.66$
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:  $19, 1129$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{33} a^{16} + \frac{13}{33} a^{15} + \frac{2}{33} a^{14} - \frac{14}{33} a^{13} - \frac{16}{33} a^{12} + \frac{7}{33} a^{11} - \frac{16}{33} a^{10} - \frac{2}{33} a^{9} + \frac{1}{3} a^{8} - \frac{4}{33} a^{7} - \frac{7}{33} a^{6} + \frac{14}{33} a^{5} - \frac{3}{11} a^{4} - \frac{13}{33} a^{3} + \frac{16}{33} a^{2} - \frac{1}{3} a + \frac{13}{33}$, $\frac{1}{528888496857712284763239216989503436996841250029} a^{17} + \frac{10256144438782548360979621017362885261396117}{10370362683484554603200768960578498764643946079} a^{16} - \frac{197989899765099425186073413893193177027541653963}{528888496857712284763239216989503436996841250029} a^{15} - \frac{66311783734581211079396266822736243571913295602}{528888496857712284763239216989503436996841250029} a^{14} - \frac{50389177696709936047031699128397776401201405413}{528888496857712284763239216989503436996841250029} a^{13} - \frac{90200611383640392947928655229334054857661504259}{528888496857712284763239216989503436996841250029} a^{12} + \frac{69709829772315930329395693771987755705571461634}{528888496857712284763239216989503436996841250029} a^{11} - \frac{252980319936041968018409931869612399681373132643}{528888496857712284763239216989503436996841250029} a^{10} + \frac{251991506194729217671752908415934544866505044966}{528888496857712284763239216989503436996841250029} a^{9} - \frac{17312658328501283845320171673010766373379373640}{176296165619237428254413072329834478998947083343} a^{8} + \frac{68854051795334291775608214447103803293185504340}{176296165619237428254413072329834478998947083343} a^{7} - \frac{71735323816928741418093685776405282181395340520}{176296165619237428254413072329834478998947083343} a^{6} + \frac{12474455694691496713601693947762498401317909855}{48080772441610207705749019726318494272440113639} a^{5} - \frac{102713713659220641267081664824894550576534138921}{528888496857712284763239216989503436996841250029} a^{4} - \frac{20497099662769303582027629378452734459850778184}{528888496857712284763239216989503436996841250029} a^{3} + \frac{21871864910490210868875234692407057468006779328}{176296165619237428254413072329834478998947083343} a^{2} + \frac{2604782455435403624692133812041577655567639113}{58765388539745809418137690776611492999649027781} a - \frac{3404938893821648863862312939208165050890476662}{31111088050453663809602306881735496293931838237}$

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:  $17$
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:  \( 367625216248 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_9$ (as 18T39):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $C_2^2:D_9$
Character table for $C_2^2:D_9$

Intermediate fields

3.3.1129.1, 6.6.519504157729.1, 9.9.1624709678881.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
1129Data not computed