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Properties

Label 18.0.64809941654...5312.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 7^{9}\cdot 11^{6}\cdot 19^{12}$
Root discriminant $66.51$
Ramified primes $2, 7, 11, 19$
Class number $3536$ (GRH)
Class group $[2, 2, 884]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![869143, 990627, 783486, 401368, 410031, 301625, 153456, 34739, 19966, 14595, 4834, -2013, -1018, 227, 269, 14, -18, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 18*x^16 + 14*x^15 + 269*x^14 + 227*x^13 - 1018*x^12 - 2013*x^11 + 4834*x^10 + 14595*x^9 + 19966*x^8 + 34739*x^7 + 153456*x^6 + 301625*x^5 + 410031*x^4 + 401368*x^3 + 783486*x^2 + 990627*x + 869143)
 
gp: K = bnfinit(x^18 - 3*x^17 - 18*x^16 + 14*x^15 + 269*x^14 + 227*x^13 - 1018*x^12 - 2013*x^11 + 4834*x^10 + 14595*x^9 + 19966*x^8 + 34739*x^7 + 153456*x^6 + 301625*x^5 + 410031*x^4 + 401368*x^3 + 783486*x^2 + 990627*x + 869143, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 18 x^{16} + 14 x^{15} + 269 x^{14} + 227 x^{13} - 1018 x^{12} - 2013 x^{11} + 4834 x^{10} + 14595 x^{9} + 19966 x^{8} + 34739 x^{7} + 153456 x^{6} + 301625 x^{5} + 410031 x^{4} + 401368 x^{3} + 783486 x^{2} + 990627 x + 869143 \)

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:  \(-648099416547998746795472637325312=-\,2^{12}\cdot 7^{9}\cdot 11^{6}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.51$
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:  $2, 7, 11, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{22} a^{14} + \frac{2}{11} a^{13} - \frac{5}{22} a^{12} - \frac{2}{11} a^{11} - \frac{3}{22} a^{10} + \frac{1}{11} a^{9} + \frac{9}{22} a^{8} - \frac{9}{22} a^{6} - \frac{5}{11} a^{5} - \frac{1}{2} a^{4} + \frac{4}{11} a^{3} + \frac{7}{22} a^{2}$, $\frac{1}{22} a^{15} + \frac{1}{22} a^{13} + \frac{5}{22} a^{12} - \frac{9}{22} a^{11} + \frac{3}{22} a^{10} + \frac{1}{22} a^{9} - \frac{3}{22} a^{8} - \frac{9}{22} a^{7} - \frac{7}{22} a^{6} + \frac{7}{22} a^{5} - \frac{3}{22} a^{4} - \frac{3}{22} a^{3} + \frac{5}{22} a^{2} - \frac{1}{2}$, $\frac{1}{242} a^{16} - \frac{5}{242} a^{15} + \frac{3}{242} a^{14} - \frac{47}{242} a^{13} + \frac{3}{22} a^{12} + \frac{7}{242} a^{11} + \frac{57}{242} a^{10} + \frac{7}{242} a^{9} - \frac{119}{242} a^{8} + \frac{27}{242} a^{7} - \frac{31}{242} a^{6} - \frac{113}{242} a^{5} - \frac{87}{242} a^{4} + \frac{113}{242} a^{3} + \frac{5}{11} a^{2} + \frac{3}{11} a$, $\frac{1}{1257829340226774315811246409135973949798} a^{17} - \frac{195866627850327270357682401249080321}{628914670113387157905623204567986974899} a^{16} + \frac{12246387422289989130300465161710594139}{1257829340226774315811246409135973949798} a^{15} + \frac{13844296766834161584661313227643251770}{628914670113387157905623204567986974899} a^{14} + \frac{187744139318415661646185994025909159}{10395283803527060461249970323437801238} a^{13} + \frac{213915308972549211353433370335390179297}{1257829340226774315811246409135973949798} a^{12} - \frac{424793344123153006549191921530675886803}{1257829340226774315811246409135973949798} a^{11} + \frac{7256787098130832573834540736262316967}{1257829340226774315811246409135973949798} a^{10} - \frac{202744452241906884115384309706662459737}{1257829340226774315811246409135973949798} a^{9} - \frac{330313533227759751310863267819369171021}{1257829340226774315811246409135973949798} a^{8} - \frac{283518101064456147303865563740326078917}{1257829340226774315811246409135973949798} a^{7} + \frac{40090482533008356685483306781219997529}{1257829340226774315811246409135973949798} a^{6} - \frac{163400253893926434338621189134108641287}{1257829340226774315811246409135973949798} a^{5} - \frac{19873395766351117070378385060279641409}{1257829340226774315811246409135973949798} a^{4} + \frac{359858688099281047888341066492243439}{57174060919398832536874836778907906809} a^{3} - \frac{24816822064013054883770804934652054321}{114348121838797665073749673557815813618} a^{2} + \frac{939365587615338213850813657877146282}{5197641901763530230624985161718900619} a - \frac{2390556020672372357974150272121447611}{10395283803527060461249970323437801238}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{884}$, which has order $3536$ (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:  \( -1 \) (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:  \( 654963.5034721284 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.3.15884.1, 3.3.361.1, 6.0.86539399408.2, 6.0.44700103.1, 9.9.4007556327104.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
19Data not computed