↑

Properties

Label 18.6.20965741778...0672.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 13^{15}$
Root discriminant $13.46$
Ramified primes $2, 13$
Class number $1$
Class group Trivial
Galois group $S_3 \times C_6$ (as 18T6)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 6, -3, 8, -39, 35, -12, 34, -59, 54, -33, 13, -14, 16, -5, -6, 9, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 9*x^16 - 6*x^15 - 5*x^14 + 16*x^13 - 14*x^12 + 13*x^11 - 33*x^10 + 54*x^9 - 59*x^8 + 34*x^7 - 12*x^6 + 35*x^5 - 39*x^4 + 8*x^3 - 3*x^2 + 6*x - 1)
 
gp: K = bnfinit(x^18 - 5*x^17 + 9*x^16 - 6*x^15 - 5*x^14 + 16*x^13 - 14*x^12 + 13*x^11 - 33*x^10 + 54*x^9 - 59*x^8 + 34*x^7 - 12*x^6 + 35*x^5 - 39*x^4 + 8*x^3 - 3*x^2 + 6*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 9 x^{16} - 6 x^{15} - 5 x^{14} + 16 x^{13} - 14 x^{12} + 13 x^{11} - 33 x^{10} + 54 x^{9} - 59 x^{8} + 34 x^{7} - 12 x^{6} + 35 x^{5} - 39 x^{4} + 8 x^{3} - 3 x^{2} + 6 x - 1 \)

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:  \(209657417785715740672=2^{12}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13.46$
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:  $2, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{233} a^{16} + \frac{55}{233} a^{15} - \frac{99}{233} a^{14} + \frac{4}{233} a^{13} + \frac{10}{233} a^{12} + \frac{32}{233} a^{11} - \frac{20}{233} a^{10} - \frac{34}{233} a^{9} - \frac{85}{233} a^{8} - \frac{82}{233} a^{7} - \frac{25}{233} a^{6} + \frac{21}{233} a^{5} + \frac{5}{233} a^{4} + \frac{65}{233} a^{3} + \frac{102}{233} a^{2} - \frac{100}{233} a + \frac{75}{233}$, $\frac{1}{12903307} a^{17} + \frac{14522}{12903307} a^{16} + \frac{22492}{12903307} a^{15} + \frac{2607991}{12903307} a^{14} - \frac{5355644}{12903307} a^{13} + \frac{461349}{12903307} a^{12} - \frac{946493}{12903307} a^{11} + \frac{3615473}{12903307} a^{10} + \frac{4907579}{12903307} a^{9} - \frac{696906}{12903307} a^{8} - \frac{672321}{12903307} a^{7} - \frac{5372319}{12903307} a^{6} + \frac{4438630}{12903307} a^{5} - \frac{1723564}{12903307} a^{4} - \frac{2420568}{12903307} a^{3} + \frac{3125873}{12903307} a^{2} - \frac{3105818}{12903307} a + \frac{1600654}{12903307}$

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

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1074.07507043 \)
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

\(\Q(\sqrt{13}) \), 3.3.169.1, 3.1.676.1, \(\Q(\zeta_{13})^+\), 6.2.5940688.1, 9.3.308915776.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
13Data not computed