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Properties

Label 18.18.7070897091...0625.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{27}\cdot 5^{9}\cdot 7^{15}$
Root discriminant $58.81$
Ramified primes $3, 5, 7$
Class number $2$ (GRH)
Class group $[2]$ (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![-2456, 83436, 775098, -152903, -1705209, 97803, 1414748, -37629, -588942, 8571, 137472, -1035, -18690, 57, 1458, -1, -60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 60*x^16 - x^15 + 1458*x^14 + 57*x^13 - 18690*x^12 - 1035*x^11 + 137472*x^10 + 8571*x^9 - 588942*x^8 - 37629*x^7 + 1414748*x^6 + 97803*x^5 - 1705209*x^4 - 152903*x^3 + 775098*x^2 + 83436*x - 2456)
 
gp: K = bnfinit(x^18 - 60*x^16 - x^15 + 1458*x^14 + 57*x^13 - 18690*x^12 - 1035*x^11 + 137472*x^10 + 8571*x^9 - 588942*x^8 - 37629*x^7 + 1414748*x^6 + 97803*x^5 - 1705209*x^4 - 152903*x^3 + 775098*x^2 + 83436*x - 2456, 1)
 

Normalized defining polynomial

\( x^{18} - 60 x^{16} - x^{15} + 1458 x^{14} + 57 x^{13} - 18690 x^{12} - 1035 x^{11} + 137472 x^{10} + 8571 x^{9} - 588942 x^{8} - 37629 x^{7} + 1414748 x^{6} + 97803 x^{5} - 1705209 x^{4} - 152903 x^{3} + 775098 x^{2} + 83436 x - 2456 \)

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:  \(70708970918051611315870587890625=3^{27}\cdot 5^{9}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.81$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(256,·)$, $\chi_{315}(1,·)$, $\chi_{315}(194,·)$, $\chi_{315}(269,·)$, $\chi_{315}(16,·)$, $\chi_{315}(209,·)$, $\chi_{315}(211,·)$, $\chi_{315}(151,·)$, $\chi_{315}(89,·)$, $\chi_{315}(226,·)$, $\chi_{315}(164,·)$, $\chi_{315}(104,·)$, $\chi_{315}(106,·)$, $\chi_{315}(299,·)$, $\chi_{315}(46,·)$, $\chi_{315}(121,·)$, $\chi_{315}(314,·)$, $\chi_{315}(59,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{52} a^{15} + \frac{3}{26} a^{13} - \frac{1}{26} a^{12} + \frac{2}{13} a^{11} + \frac{5}{26} a^{10} - \frac{3}{13} a^{9} - \frac{3}{26} a^{8} - \frac{3}{13} a^{7} - \frac{4}{13} a^{6} + \frac{1}{13} a^{5} + \frac{1}{26} a^{4} + \frac{1}{13} a^{3} + \frac{1}{26} a^{2} + \frac{3}{52} a + \frac{7}{26}$, $\frac{1}{65468} a^{16} - \frac{77}{65468} a^{15} - \frac{3611}{32734} a^{14} + \frac{1681}{65468} a^{13} - \frac{1500}{16367} a^{12} - \frac{1165}{65468} a^{11} - \frac{2828}{16367} a^{10} + \frac{4545}{65468} a^{9} - \frac{6509}{32734} a^{8} - \frac{15901}{65468} a^{7} - \frac{4397}{16367} a^{6} + \frac{20585}{65468} a^{5} + \frac{12977}{32734} a^{4} - \frac{27879}{65468} a^{3} - \frac{437}{65468} a^{2} + \frac{2614}{16367} a - \frac{926}{16367}$, $\frac{1}{345334917061067119716649039856} a^{17} + \frac{665474350130746608959631}{172667458530533559858324519928} a^{16} - \frac{30408926208275904969034794}{21583432316316694982290564991} a^{15} + \frac{22967876107728282198431294335}{345334917061067119716649039856} a^{14} + \frac{3921872638082394301228053095}{43166864632633389964581129982} a^{13} + \frac{34617975947745129027417064345}{345334917061067119716649039856} a^{12} + \frac{4113929989637476548654159048}{21583432316316694982290564991} a^{11} + \frac{86122768002659511800683756749}{345334917061067119716649039856} a^{10} - \frac{12263581643671105901738045267}{172667458530533559858324519928} a^{9} - \frac{15189298302682956974991909265}{345334917061067119716649039856} a^{8} - \frac{5799468352961610348360338707}{43166864632633389964581129982} a^{7} + \frac{34420073568047180237394609475}{345334917061067119716649039856} a^{6} - \frac{90745941701635872003483601}{455586961821988284586608232} a^{5} + \frac{52149001846165159495279955767}{345334917061067119716649039856} a^{4} + \frac{45396021563057010140003581669}{345334917061067119716649039856} a^{3} + \frac{85287823107405117388053173215}{345334917061067119716649039856} a^{2} + \frac{5481499540388894329508565506}{21583432316316694982290564991} a - \frac{37762169895384127163147411789}{86333729265266779929162259964}$

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:  \( 9362411700.9 \) (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

\(\Q(\sqrt{105}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 3.3.3969.1, 6.6.843908625.1, 6.6.41351522625.2, 6.6.56723625.1, 6.6.41351522625.1, 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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed