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Properties

Label 18.6.29887202898...6013.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{24}\cdot 7^{12}\cdot 197^{3}$
Root discriminant $38.19$
Ramified primes $3, 7, 197$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T263

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-631, 6405, 7359, -12980, -7617, 7392, -7941, -5754, 6816, -1626, -477, 1857, -1152, 531, -120, -12, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 18*x^16 - 12*x^15 - 120*x^14 + 531*x^13 - 1152*x^12 + 1857*x^11 - 477*x^10 - 1626*x^9 + 6816*x^8 - 5754*x^7 - 7941*x^6 + 7392*x^5 - 7617*x^4 - 12980*x^3 + 7359*x^2 + 6405*x - 631)
 
gp: K = bnfinit(x^18 - 6*x^17 + 18*x^16 - 12*x^15 - 120*x^14 + 531*x^13 - 1152*x^12 + 1857*x^11 - 477*x^10 - 1626*x^9 + 6816*x^8 - 5754*x^7 - 7941*x^6 + 7392*x^5 - 7617*x^4 - 12980*x^3 + 7359*x^2 + 6405*x - 631, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 18 x^{16} - 12 x^{15} - 120 x^{14} + 531 x^{13} - 1152 x^{12} + 1857 x^{11} - 477 x^{10} - 1626 x^{9} + 6816 x^{8} - 5754 x^{7} - 7941 x^{6} + 7392 x^{5} - 7617 x^{4} - 12980 x^{3} + 7359 x^{2} + 6405 x - 631 \)

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:  \(29887202898467161742960366013=3^{24}\cdot 7^{12}\cdot 197^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.19$
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:  $3, 7, 197$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2369954526568853616576444027767697175858} a^{17} - \frac{107249285085427940757355232644576213266}{1184977263284426808288222013883848587929} a^{16} - \frac{412507694695514646463432466488457013885}{2369954526568853616576444027767697175858} a^{15} - \frac{280750317010977744951293736911570469459}{1184977263284426808288222013883848587929} a^{14} + \frac{99258417501372482815768723863294423327}{1184977263284426808288222013883848587929} a^{13} - \frac{259733457690804818418408451051432343927}{2369954526568853616576444027767697175858} a^{12} - \frac{667612178836933659044767220089407996247}{2369954526568853616576444027767697175858} a^{11} + \frac{1076482020928770640838469315376913406831}{2369954526568853616576444027767697175858} a^{10} + \frac{1041009343597183274838386331547282348989}{2369954526568853616576444027767697175858} a^{9} + \frac{159867409986295284031076063357078106141}{2369954526568853616576444027767697175858} a^{8} + \frac{501476299659317302454840566684106376481}{2369954526568853616576444027767697175858} a^{7} - \frac{127654391864727232205189845813152382193}{2369954526568853616576444027767697175858} a^{6} - \frac{709754429794075898215537712793160679913}{2369954526568853616576444027767697175858} a^{5} - \frac{648725429772388665688835698449617417567}{2369954526568853616576444027767697175858} a^{4} - \frac{117950058628873542669753853635296303057}{1184977263284426808288222013883848587929} a^{3} + \frac{887610973388289228105267487310398554581}{2369954526568853616576444027767697175858} a^{2} + \frac{393355538741278643516571117511965841872}{1184977263284426808288222013883848587929} a + \frac{313774626266170880997477894236508927981}{1184977263284426808288222013883848587929}$

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:  \( 12743863.8491 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T263:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1152
The 48 conjugacy class representatives for t18n263
Character table for t18n263 is not computed

Intermediate fields

3.3.3969.2, 3.3.3969.1, \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 9.9.62523502209.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
197Data not computed