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Properties

Label 18.6.29531293631...7077.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{19}\cdot 31^{11}$
Root discriminant $26.00$
Ramified primes $3, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times S_4$ (as 18T65)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 39, -36, 120, -687, 329, 1185, -72, -942, -696, 66, 324, 108, -27, -31, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^16 - 31*x^15 - 27*x^14 + 108*x^13 + 324*x^12 + 66*x^11 - 696*x^10 - 942*x^9 - 72*x^8 + 1185*x^7 + 329*x^6 - 687*x^5 + 120*x^4 - 36*x^3 + 39*x^2 - 3)
 
gp: K = bnfinit(x^18 - 9*x^16 - 31*x^15 - 27*x^14 + 108*x^13 + 324*x^12 + 66*x^11 - 696*x^10 - 942*x^9 - 72*x^8 + 1185*x^7 + 329*x^6 - 687*x^5 + 120*x^4 - 36*x^3 + 39*x^2 - 3, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{16} - 31 x^{15} - 27 x^{14} + 108 x^{13} + 324 x^{12} + 66 x^{11} - 696 x^{10} - 942 x^{9} - 72 x^{8} + 1185 x^{7} + 329 x^{6} - 687 x^{5} + 120 x^{4} - 36 x^{3} + 39 x^{2} - 3 \)

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:  \(29531293631851085903947077=3^{19}\cdot 31^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.00$
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:  $3, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{7} a^{14} + \frac{2}{7} a^{13} - \frac{2}{7} a^{12} + \frac{2}{7} a^{11} + \frac{1}{7} a^{10} - \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} + \frac{2}{7} a^{5} + \frac{2}{7} a^{3} + \frac{1}{7} a^{2} - \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{15} + \frac{1}{7} a^{13} - \frac{1}{7} a^{12} - \frac{3}{7} a^{11} + \frac{2}{7} a^{10} - \frac{2}{7} a^{9} + \frac{3}{7} a^{8} + \frac{2}{7} a^{7} + \frac{1}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{3}{7} a^{3} - \frac{3}{7} a^{2} + \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{49} a^{16} + \frac{1}{49} a^{15} + \frac{1}{49} a^{14} - \frac{3}{7} a^{13} + \frac{10}{49} a^{12} - \frac{22}{49} a^{11} + \frac{15}{49} a^{9} + \frac{5}{49} a^{8} - \frac{18}{49} a^{7} + \frac{4}{49} a^{6} - \frac{2}{49} a^{5} + \frac{20}{49} a^{4} + \frac{1}{49} a^{3} + \frac{1}{7} a^{2} - \frac{20}{49} a - \frac{16}{49}$, $\frac{1}{679143895114991761} a^{17} - \frac{217997220553277}{679143895114991761} a^{16} - \frac{26402311184042371}{679143895114991761} a^{15} + \frac{2019108788049740}{679143895114991761} a^{14} - \frac{15817371930147637}{52241838085768597} a^{13} - \frac{156170061924908495}{679143895114991761} a^{12} - \frac{75461285061876933}{679143895114991761} a^{11} + \frac{252721866203012956}{679143895114991761} a^{10} - \frac{13415605361589035}{97020556444998823} a^{9} + \frac{71903499429443051}{679143895114991761} a^{8} + \frac{52894564884261663}{679143895114991761} a^{7} - \frac{315169219232623400}{679143895114991761} a^{6} + \frac{253484604891211458}{679143895114991761} a^{5} + \frac{235692668818077036}{679143895114991761} a^{4} + \frac{77266854029732593}{679143895114991761} a^{3} - \frac{161722173082731134}{679143895114991761} a^{2} - \frac{38871069190915713}{97020556444998823} a - \frac{186298145363003020}{679143895114991761}$

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:  \( 782356.481444 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times S_4$ (as 18T65):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 15 conjugacy class representatives for $S_3\times S_4$
Character table for $S_3\times S_4$

Intermediate fields

3.3.837.1, 3.1.31.1, 6.2.89373.1, 9.3.18177663843.1

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

Sibling fields

Degree 12 sibling: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.6.7.5$x^{6} + 6 x^{2} + 3$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
3.12.12.5$x^{12} + 33 x^{11} - 63 x^{10} - 36 x^{9} - 90 x^{8} - 54 x^{7} - 54 x^{6} - 108 x^{4} - 27 x^{3} - 81 x^{2} + 81 x - 81$$3$$4$$12$$S_3 \times C_4$$[3/2]_{2}^{4}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.2.1$x^{4} + 713 x^{2} + 138384$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31.4.3.1$x^{4} + 217$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$