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Properties

Label 18.6.19831037771...2656.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{8}\cdot 3^{14}\cdot 503^{6}$
Root discriminant $25.43$
Ramified primes $2, 3, 503$
Class number $1$
Class group Trivial
Galois group $(C_3\times A_4):S_3$ (as 18T108)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 47, -270, 188, 897, -659, -1501, 637, 1359, 19, -369, -280, 4, 83, 27, -10, -3, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 3*x^16 - 10*x^15 + 27*x^14 + 83*x^13 + 4*x^12 - 280*x^11 - 369*x^10 + 19*x^9 + 1359*x^8 + 637*x^7 - 1501*x^6 - 659*x^5 + 897*x^4 + 188*x^3 - 270*x^2 + 47*x - 1)
 
gp: K = bnfinit(x^18 - 2*x^17 - 3*x^16 - 10*x^15 + 27*x^14 + 83*x^13 + 4*x^12 - 280*x^11 - 369*x^10 + 19*x^9 + 1359*x^8 + 637*x^7 - 1501*x^6 - 659*x^5 + 897*x^4 + 188*x^3 - 270*x^2 + 47*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 3 x^{16} - 10 x^{15} + 27 x^{14} + 83 x^{13} + 4 x^{12} - 280 x^{11} - 369 x^{10} + 19 x^{9} + 1359 x^{8} + 637 x^{7} - 1501 x^{6} - 659 x^{5} + 897 x^{4} + 188 x^{3} - 270 x^{2} + 47 x - 1 \)

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:  \(19831037771561498863462656=2^{8}\cdot 3^{14}\cdot 503^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.43$
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:  $2, 3, 503$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{4}{11} a^{13} + \frac{3}{11} a^{12} - \frac{3}{11} a^{11} + \frac{2}{11} a^{10} - \frac{1}{11} a^{9} + \frac{5}{11} a^{8} + \frac{1}{11} a^{7} + \frac{3}{11} a^{5} + \frac{2}{11} a^{4} - \frac{5}{11} a^{3} + \frac{5}{11} a^{2} - \frac{4}{11} a + \frac{2}{11}$, $\frac{1}{55} a^{15} + \frac{2}{55} a^{14} + \frac{1}{55} a^{13} + \frac{3}{11} a^{12} + \frac{6}{55} a^{11} + \frac{1}{5} a^{10} - \frac{1}{55} a^{9} - \frac{24}{55} a^{8} - \frac{27}{55} a^{7} + \frac{3}{55} a^{6} - \frac{2}{55} a^{5} - \frac{26}{55} a^{4} - \frac{5}{11} a^{3} - \frac{18}{55} a^{2} + \frac{1}{5} a - \frac{21}{55}$, $\frac{1}{9845} a^{16} + \frac{51}{9845} a^{15} - \frac{16}{9845} a^{14} + \frac{2394}{9845} a^{13} - \frac{69}{895} a^{12} - \frac{783}{1969} a^{11} - \frac{207}{895} a^{10} - \frac{2818}{9845} a^{9} - \frac{18}{9845} a^{8} + \frac{208}{1969} a^{7} + \frac{381}{1969} a^{6} + \frac{4206}{9845} a^{5} + \frac{131}{895} a^{4} + \frac{102}{9845} a^{3} + \frac{3339}{9845} a^{2} - \frac{2322}{9845} a + \frac{2096}{9845}$, $\frac{1}{704317905495179195} a^{17} - \frac{31298490284614}{704317905495179195} a^{16} + \frac{82596573846669}{64028900499561745} a^{15} + \frac{31243048374541279}{704317905495179195} a^{14} - \frac{225349459528699744}{704317905495179195} a^{13} + \frac{63255098544682117}{140863581099035839} a^{12} - \frac{116728651561666527}{704317905495179195} a^{11} + \frac{54725511478815742}{704317905495179195} a^{10} + \frac{114913848348224872}{704317905495179195} a^{9} + \frac{64512996104104347}{140863581099035839} a^{8} - \frac{5153569931960679}{140863581099035839} a^{7} + \frac{4116889234124421}{64028900499561745} a^{6} - \frac{299192518412412784}{704317905495179195} a^{5} + \frac{64575664400716552}{704317905495179195} a^{4} + \frac{2518973460205874}{5820809136323795} a^{3} - \frac{245315834496764052}{704317905495179195} a^{2} + \frac{53138678566359676}{704317905495179195} a - \frac{31442530569350833}{140863581099035839}$

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:  \( 665732.430448 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_3\times A_4):S_3$ (as 18T108):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 216
The 19 conjugacy class representatives for $(C_3\times A_4):S_3$
Character table for $(C_3\times A_4):S_3$

Intermediate fields

3.3.1509.1, 6.2.2277081.1, 9.9.4453205336784.1

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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.8.2$x^{12} - 8 x^{3} + 16$$3$$4$$8$$C_3\times (C_3 : C_4)$$[\ ]_{3}^{12}$
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.12.14.15$x^{12} + 9 x^{11} - 6 x^{10} + 6 x^{9} - 3 x^{8} + 9 x^{7} + 6 x^{6} - 9 x^{5} - 9 x^{4} - 9 x^{3} - 9 x^{2} + 9$$6$$2$$14$$C_6\times S_3$$[3/2]_{2}^{6}$
503Data not computed