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Properties

Label 18.10.8559811674...8992.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{12}\cdot 37^{9}\cdot 401^{2}$
Root discriminant $18.79$
Ramified primes $2, 37, 401$
Class number $1$
Class group Trivial
Galois group 18T319

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, 0, 45, -79, 93, 53, -170, -151, 273, 76, -216, 72, 41, -46, 14, 3, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 3*x^16 + 14*x^15 - 46*x^14 + 41*x^13 + 72*x^12 - 216*x^11 + 76*x^10 + 273*x^9 - 151*x^8 - 170*x^7 + 53*x^6 + 93*x^5 - 79*x^4 + 45*x^3 - 7*x + 1)
 
gp: K = bnfinit(x^18 - 4*x^17 + 3*x^16 + 14*x^15 - 46*x^14 + 41*x^13 + 72*x^12 - 216*x^11 + 76*x^10 + 273*x^9 - 151*x^8 - 170*x^7 + 53*x^6 + 93*x^5 - 79*x^4 + 45*x^3 - 7*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 3 x^{16} + 14 x^{15} - 46 x^{14} + 41 x^{13} + 72 x^{12} - 216 x^{11} + 76 x^{10} + 273 x^{9} - 151 x^{8} - 170 x^{7} + 53 x^{6} + 93 x^{5} - 79 x^{4} + 45 x^{3} - 7 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(85598116744348371668992=2^{12}\cdot 37^{9}\cdot 401^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.79$
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:  $2, 37, 401$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} - \frac{1}{5} a^{14} - \frac{1}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{16} - \frac{2}{5} a^{14} - \frac{2}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{4} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{7831639679363015} a^{17} + \frac{223456943777}{2886708322655} a^{16} - \frac{768786238887446}{7831639679363015} a^{15} - \frac{629720754714886}{1566327935872603} a^{14} + \frac{3469230569999861}{7831639679363015} a^{13} - \frac{2313574549491022}{7831639679363015} a^{12} - \frac{62258210152357}{1566327935872603} a^{11} + \frac{710283427365014}{7831639679363015} a^{10} - \frac{3409231920427309}{7831639679363015} a^{9} - \frac{4568652095646}{26729145663355} a^{8} - \frac{724366957973718}{1566327935872603} a^{7} + \frac{2633771916595714}{7831639679363015} a^{6} - \frac{865670002692898}{7831639679363015} a^{5} + \frac{1632898211173816}{7831639679363015} a^{4} + \frac{2113469208138912}{7831639679363015} a^{3} + \frac{2928878401997872}{7831639679363015} a^{2} + \frac{981955458301798}{7831639679363015} a - \frac{1529419352056806}{7831639679363015}$

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:  $13$
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:  \( 57675.2080079 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T319:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1296
The 22 conjugacy class representatives for t18n319
Character table for t18n319 is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.148.1 x3, 6.6.810448.1, 9.5.1299958592.1

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

Sibling fields

Degree 9 sibling: data not computed
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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
2Data not computed
37Data not computed
401Data not computed