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Properties

Label 18.0.20621522147...6944.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 7^{15}\cdot 13^{9}$
Root discriminant $28.97$
Ramified primes $2, 7, 13$
Class number $8$
Class group $[2, 2, 2]$
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![533, -1664, 3422, -5616, 7994, -8995, 9045, -8162, 6890, -5418, 4197, -2898, 1709, -742, 294, -81, 27, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 27*x^16 - 81*x^15 + 294*x^14 - 742*x^13 + 1709*x^12 - 2898*x^11 + 4197*x^10 - 5418*x^9 + 6890*x^8 - 8162*x^7 + 9045*x^6 - 8995*x^5 + 7994*x^4 - 5616*x^3 + 3422*x^2 - 1664*x + 533)
 
gp: K = bnfinit(x^18 - 3*x^17 + 27*x^16 - 81*x^15 + 294*x^14 - 742*x^13 + 1709*x^12 - 2898*x^11 + 4197*x^10 - 5418*x^9 + 6890*x^8 - 8162*x^7 + 9045*x^6 - 8995*x^5 + 7994*x^4 - 5616*x^3 + 3422*x^2 - 1664*x + 533, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 27 x^{16} - 81 x^{15} + 294 x^{14} - 742 x^{13} + 1709 x^{12} - 2898 x^{11} + 4197 x^{10} - 5418 x^{9} + 6890 x^{8} - 8162 x^{7} + 9045 x^{6} - 8995 x^{5} + 7994 x^{4} - 5616 x^{3} + 3422 x^{2} - 1664 x + 533 \)

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:  \(-206215221475202976832466944=-\,2^{12}\cdot 7^{15}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.97$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{2}{13} a^{13} - \frac{5}{13} a^{11} - \frac{6}{13} a^{10} + \frac{5}{13} a^{9} - \frac{3}{13} a^{8} - \frac{5}{13} a^{7} - \frac{6}{13} a^{6} + \frac{5}{13} a^{5} - \frac{2}{13} a^{3} + \frac{1}{13} a^{2}$, $\frac{1}{299} a^{15} + \frac{10}{299} a^{14} - \frac{62}{299} a^{13} + \frac{21}{299} a^{12} + \frac{45}{299} a^{11} + \frac{48}{299} a^{10} + \frac{128}{299} a^{9} - \frac{55}{299} a^{8} - \frac{2}{13} a^{7} - \frac{134}{299} a^{6} + \frac{105}{299} a^{5} - \frac{106}{299} a^{4} - \frac{15}{299} a^{3} - \frac{5}{299} a^{2} + \frac{5}{23} a - \frac{4}{23}$, $\frac{1}{3887} a^{16} + \frac{6}{3887} a^{15} - \frac{79}{3887} a^{14} + \frac{614}{3887} a^{13} - \frac{49}{299} a^{12} + \frac{96}{299} a^{11} + \frac{1293}{3887} a^{10} + \frac{1043}{3887} a^{9} + \frac{1899}{3887} a^{8} - \frac{5}{299} a^{7} - \frac{95}{3887} a^{6} + \frac{1383}{3887} a^{5} - \frac{488}{3887} a^{4} - \frac{1785}{3887} a^{3} + \frac{1902}{3887} a^{2} + \frac{7}{23} a + \frac{85}{299}$, $\frac{1}{2756658052158390888287117} a^{17} + \frac{71752212054795046782}{2756658052158390888287117} a^{16} - \frac{2003710825491272518421}{2756658052158390888287117} a^{15} - \frac{65950777358299667264598}{2756658052158390888287117} a^{14} + \frac{970612244689404011386877}{2756658052158390888287117} a^{13} - \frac{83969675792269001008035}{212050619396799299099009} a^{12} + \frac{419804089303224302019089}{2756658052158390888287117} a^{11} - \frac{526997114349401056654401}{2756658052158390888287117} a^{10} - \frac{777565836898672088391250}{2756658052158390888287117} a^{9} + \frac{734483292328018676563849}{2756658052158390888287117} a^{8} + \frac{413172601333034620218670}{2756658052158390888287117} a^{7} - \frac{1270433280484686021318159}{2756658052158390888287117} a^{6} + \frac{882601288685294764303599}{2756658052158390888287117} a^{5} + \frac{566511150490487397688820}{2756658052158390888287117} a^{4} - \frac{60934577092370916703155}{212050619396799299099009} a^{3} - \frac{1169219265542353416743127}{2756658052158390888287117} a^{2} + \frac{15507499844914936818434}{212050619396799299099009} a - \frac{2300494090525028866614}{5171966326751202417049}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 140111.524592 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-91}) \), 3.1.364.1 x3, \(\Q(\zeta_{7})^+\), 6.0.12057136.1, 6.0.590799664.2 x2, 6.0.36924979.1, 9.3.115796734144.2 x3

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

Sibling fields

Degree 6 sibling: 6.0.590799664.2
Degree 9 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
2Data not computed
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$