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Properties

Label 18.6.36644198070...0625.2
Degree $18$
Signature $[6, 6]$
Discriminant $3^{36}\cdot 5^{12}$
Root discriminant $26.32$
Ramified primes $3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2:S_3.C_2$ (as 18T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, -252, 576, -1026, 1269, -1116, 840, -270, 135, -14, -261, 72, -18, 54, 9, -12, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^15 + 9*x^14 + 54*x^13 - 18*x^12 + 72*x^11 - 261*x^10 - 14*x^9 + 135*x^8 - 270*x^7 + 840*x^6 - 1116*x^5 + 1269*x^4 - 1026*x^3 + 576*x^2 - 252*x + 49)
 
gp: K = bnfinit(x^18 - 12*x^15 + 9*x^14 + 54*x^13 - 18*x^12 + 72*x^11 - 261*x^10 - 14*x^9 + 135*x^8 - 270*x^7 + 840*x^6 - 1116*x^5 + 1269*x^4 - 1026*x^3 + 576*x^2 - 252*x + 49, 1)
 

Normalized defining polynomial

\( x^{18} - 12 x^{15} + 9 x^{14} + 54 x^{13} - 18 x^{12} + 72 x^{11} - 261 x^{10} - 14 x^{9} + 135 x^{8} - 270 x^{7} + 840 x^{6} - 1116 x^{5} + 1269 x^{4} - 1026 x^{3} + 576 x^{2} - 252 x + 49 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(36644198070556426025390625=3^{36}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.32$
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:  $3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{6} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{18} a^{12} - \frac{1}{18} a^{9} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{9} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{18}$, $\frac{1}{18} a^{13} - \frac{1}{18} a^{10} - \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{9} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{18} a$, $\frac{1}{36} a^{14} - \frac{1}{36} a^{12} + \frac{1}{18} a^{11} - \frac{1}{12} a^{10} - \frac{1}{18} a^{9} + \frac{1}{6} a^{8} + \frac{1}{3} a^{6} + \frac{1}{18} a^{5} + \frac{1}{4} a^{4} - \frac{1}{18} a^{3} - \frac{11}{36} a^{2} - \frac{1}{6} a - \frac{7}{36}$, $\frac{1}{1260} a^{15} - \frac{23}{1260} a^{13} + \frac{4}{315} a^{12} - \frac{19}{420} a^{11} + \frac{23}{315} a^{10} - \frac{11}{315} a^{9} - \frac{23}{210} a^{7} - \frac{22}{45} a^{6} - \frac{29}{140} a^{5} + \frac{146}{315} a^{4} - \frac{65}{252} a^{3} - \frac{61}{210} a^{2} + \frac{421}{1260} a + \frac{14}{45}$, $\frac{1}{42840} a^{16} + \frac{1}{42840} a^{15} - \frac{29}{21420} a^{14} + \frac{23}{2040} a^{13} - \frac{47}{2380} a^{12} + \frac{53}{1224} a^{11} - \frac{337}{42840} a^{10} + \frac{1483}{21420} a^{9} - \frac{382}{1785} a^{8} + \frac{982}{5355} a^{7} + \frac{6263}{42840} a^{6} + \frac{7813}{42840} a^{5} - \frac{199}{510} a^{4} + \frac{5323}{14280} a^{3} - \frac{523}{4284} a^{2} - \frac{7307}{42840} a - \frac{779}{6120}$, $\frac{1}{3811174920} a^{17} + \frac{83}{22685565} a^{16} + \frac{23403}{60494840} a^{15} - \frac{17816881}{3811174920} a^{14} + \frac{104639411}{3811174920} a^{13} + \frac{60581911}{3811174920} a^{12} - \frac{2156687}{63519582} a^{11} + \frac{12077789}{224186760} a^{10} - \frac{101591183}{1905587460} a^{9} - \frac{7777268}{68056695} a^{8} - \frac{52532269}{423463880} a^{7} + \frac{121517113}{635195820} a^{6} - \frac{34492943}{544453560} a^{5} + \frac{69952747}{3811174920} a^{4} - \frac{1823215483}{3811174920} a^{3} + \frac{87347181}{423463880} a^{2} + \frac{177255977}{381117492} a - \frac{86579903}{544453560}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 1592430.98785 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3.C_2$ (as 18T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 14 conjugacy class representatives for $C_3^2:S_3.C_2$
Character table for $C_3^2:S_3.C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 6.2.4100625.2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }^{3}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
3Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$