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Properties

Label 18.8.56986226150...6707.4
Degree $18$
Signature $[8, 5]$
Discriminant $-\,3^{27}\cdot 73^{3}\cdot 577^{3}$
Root discriminant $30.65$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T767

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-557, -2958, 9411, -15632, 16947, -6912, -7305, 14874, -13560, 7335, -1827, -513, 656, -252, 54, -22, 15, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 15*x^16 - 22*x^15 + 54*x^14 - 252*x^13 + 656*x^12 - 513*x^11 - 1827*x^10 + 7335*x^9 - 13560*x^8 + 14874*x^7 - 7305*x^6 - 6912*x^5 + 16947*x^4 - 15632*x^3 + 9411*x^2 - 2958*x - 557)
 
gp: K = bnfinit(x^18 - 6*x^17 + 15*x^16 - 22*x^15 + 54*x^14 - 252*x^13 + 656*x^12 - 513*x^11 - 1827*x^10 + 7335*x^9 - 13560*x^8 + 14874*x^7 - 7305*x^6 - 6912*x^5 + 16947*x^4 - 15632*x^3 + 9411*x^2 - 2958*x - 557, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 15 x^{16} - 22 x^{15} + 54 x^{14} - 252 x^{13} + 656 x^{12} - 513 x^{11} - 1827 x^{10} + 7335 x^{9} - 13560 x^{8} + 14874 x^{7} - 7305 x^{6} - 6912 x^{5} + 16947 x^{4} - 15632 x^{3} + 9411 x^{2} - 2958 x - 557 \)

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:  \(-569862261508654647313096707=-\,3^{27}\cdot 73^{3}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.65$
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:  $3, 73, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{213256078252304710052356381423} a^{17} + \frac{102424781062306261388834815319}{213256078252304710052356381423} a^{16} - \frac{3280758241438376626483031225}{213256078252304710052356381423} a^{15} - \frac{64254672395083377910913885838}{213256078252304710052356381423} a^{14} + \frac{94876642834893941263967821063}{213256078252304710052356381423} a^{13} + \frac{16844347584864868976063823487}{213256078252304710052356381423} a^{12} + \frac{66249136030074090655936680797}{213256078252304710052356381423} a^{11} - \frac{29783117454178440373324663494}{213256078252304710052356381423} a^{10} + \frac{63770977498612964786562119668}{213256078252304710052356381423} a^{9} - \frac{27849333030465250234726953366}{213256078252304710052356381423} a^{8} - \frac{963603162296741853812400260}{213256078252304710052356381423} a^{7} + \frac{97252567512761268202731163201}{213256078252304710052356381423} a^{6} - \frac{18449260520098485664699929884}{213256078252304710052356381423} a^{5} + \frac{87701110932094244764996995953}{213256078252304710052356381423} a^{4} - \frac{12150416993985342907431798990}{213256078252304710052356381423} a^{3} - \frac{72077409869741592260543892151}{213256078252304710052356381423} a^{2} + \frac{104887968913357565713477423012}{213256078252304710052356381423} a + \frac{7311907629094856488087725269}{213256078252304710052356381423}$

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:  $12$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2873234.89067 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T767:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n767 are not computed
Character table for t18n767 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.22384826361.1

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 $18$ R $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ $18$ $18$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
3Data not computed
73Data not computed
577Data not computed