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Properties

Label 18.6.29887202898...6013.2
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![71, 162, -2457, 1442, 2862, 4443, -3033, 2994, 8115, -2843, -4875, 4443, -2006, 681, -201, 24, 12, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 12*x^16 + 24*x^15 - 201*x^14 + 681*x^13 - 2006*x^12 + 4443*x^11 - 4875*x^10 - 2843*x^9 + 8115*x^8 + 2994*x^7 - 3033*x^6 + 4443*x^5 + 2862*x^4 + 1442*x^3 - 2457*x^2 + 162*x + 71)
 
gp: K = bnfinit(x^18 - 6*x^17 + 12*x^16 + 24*x^15 - 201*x^14 + 681*x^13 - 2006*x^12 + 4443*x^11 - 4875*x^10 - 2843*x^9 + 8115*x^8 + 2994*x^7 - 3033*x^6 + 4443*x^5 + 2862*x^4 + 1442*x^3 - 2457*x^2 + 162*x + 71, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 12 x^{16} + 24 x^{15} - 201 x^{14} + 681 x^{13} - 2006 x^{12} + 4443 x^{11} - 4875 x^{10} - 2843 x^{9} + 8115 x^{8} + 2994 x^{7} - 3033 x^{6} + 4443 x^{5} + 2862 x^{4} + 1442 x^{3} - 2457 x^{2} + 162 x + 71 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{5397071232325910976620471467605720389113} a^{17} - \frac{2178715048600158306852212815209986814463}{5397071232325910976620471467605720389113} a^{16} + \frac{707087942152397379080209430532983369374}{5397071232325910976620471467605720389113} a^{15} + \frac{1569950973240870754706471963027214796408}{5397071232325910976620471467605720389113} a^{14} + \frac{988561261660098198768642926023690347852}{5397071232325910976620471467605720389113} a^{13} + \frac{1232267878502421651787154601480434333918}{5397071232325910976620471467605720389113} a^{12} + \frac{1009016520388876840778506501602409157898}{5397071232325910976620471467605720389113} a^{11} + \frac{257009718893090972603067514708420727931}{5397071232325910976620471467605720389113} a^{10} - \frac{727757436743801186947403778061322010151}{5397071232325910976620471467605720389113} a^{9} - \frac{1256794566519495396285424384851311038154}{5397071232325910976620471467605720389113} a^{8} - \frac{903221287717486977114820018815042439370}{5397071232325910976620471467605720389113} a^{7} - \frac{349910994640669755102722135257613407290}{5397071232325910976620471467605720389113} a^{6} - \frac{1831792369113066583591784715307455709234}{5397071232325910976620471467605720389113} a^{5} + \frac{2038725359843565774669508862842994839160}{5397071232325910976620471467605720389113} a^{4} + \frac{1727707526732511577394343731463714787710}{5397071232325910976620471467605720389113} a^{3} + \frac{1453130266624988319005610595952019118820}{5397071232325910976620471467605720389113} a^{2} - \frac{2326194204652565414532051753848191683221}{5397071232325910976620471467605720389113} a + \frac{2436475962571234623119267865593418862445}{5397071232325910976620471467605720389113}$

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:  \( 14123814.3583 \) (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

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