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Properties

Label 18.8.72837245972...1159.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,7^{15}\cdot 41^{2}\cdot 97^{3}$
Root discriminant $16.39$
Ramified primes $7, 41, 97$
Class number $1$
Class group Trivial
Galois group 18T472

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -5, 24, -28, 50, 1, -63, 53, -68, 6, 41, -2, 11, -13, -5, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + x^16 - 5*x^15 - 13*x^14 + 11*x^13 - 2*x^12 + 41*x^11 + 6*x^10 - 68*x^9 + 53*x^8 - 63*x^7 + x^6 + 50*x^5 - 28*x^4 + 24*x^3 - 5*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^18 + x^16 - 5*x^15 - 13*x^14 + 11*x^13 - 2*x^12 + 41*x^11 + 6*x^10 - 68*x^9 + 53*x^8 - 63*x^7 + x^6 + 50*x^5 - 28*x^4 + 24*x^3 - 5*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} + x^{16} - 5 x^{15} - 13 x^{14} + 11 x^{13} - 2 x^{12} + 41 x^{11} + 6 x^{10} - 68 x^{9} + 53 x^{8} - 63 x^{7} + x^{6} + 50 x^{5} - 28 x^{4} + 24 x^{3} - 5 x^{2} - 4 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-7283724597225833041159=-\,7^{15}\cdot 41^{2}\cdot 97^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.39$
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:  $7, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{41} a^{16} + \frac{2}{41} a^{15} - \frac{6}{41} a^{14} + \frac{2}{41} a^{13} + \frac{16}{41} a^{12} - \frac{20}{41} a^{11} - \frac{13}{41} a^{10} - \frac{11}{41} a^{9} + \frac{4}{41} a^{8} + \frac{20}{41} a^{7} + \frac{8}{41} a^{6} + \frac{20}{41} a^{5} - \frac{6}{41} a^{4} - \frac{18}{41} a^{3} + \frac{2}{41} a^{2} - \frac{20}{41} a + \frac{15}{41}$, $\frac{1}{266967626771} a^{17} + \frac{2838875090}{266967626771} a^{16} - \frac{57784793632}{266967626771} a^{15} + \frac{123164687128}{266967626771} a^{14} - \frac{1183331934}{6511405531} a^{13} + \frac{128489142008}{266967626771} a^{12} + \frac{105263897641}{266967626771} a^{11} + \frac{82740078096}{266967626771} a^{10} + \frac{64037788408}{266967626771} a^{9} + \frac{46019883696}{266967626771} a^{8} - \frac{73181569395}{266967626771} a^{7} - \frac{70734073508}{266967626771} a^{6} - \frac{22959808122}{266967626771} a^{5} + \frac{89605394250}{266967626771} a^{4} + \frac{98076145779}{266967626771} a^{3} - \frac{72768592105}{266967626771} a^{2} + \frac{121886334149}{266967626771} a + \frac{44934826173}{266967626771}$

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:  $12$
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:  \( 11002.0590961 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T472:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5184
The 88 conjugacy class representatives for t18n472 are not computed
Character table for t18n472 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.4.1630279.1, 9.5.467890073.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ $18$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
7Data not computed
41Data not computed
97Data not computed