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Properties

Label 18.6.22231110326...3216.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{18}\cdot 7^{15}\cdot 97^{3}\cdot 1399^{2}$
Root discriminant $48.52$
Ramified primes $2, 7, 97, 1399$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T472

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-313046839, 0, -48409305, 0, -17267649, 0, 1255282, 0, 350105, 0, -27440, 0, 56, 0, 210, 0, -28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 28*x^16 + 210*x^14 + 56*x^12 - 27440*x^10 + 350105*x^8 + 1255282*x^6 - 17267649*x^4 - 48409305*x^2 - 313046839)
 
gp: K = bnfinit(x^18 - 28*x^16 + 210*x^14 + 56*x^12 - 27440*x^10 + 350105*x^8 + 1255282*x^6 - 17267649*x^4 - 48409305*x^2 - 313046839, 1)
 

Normalized defining polynomial

\( x^{18} - 28 x^{16} + 210 x^{14} + 56 x^{12} - 27440 x^{10} + 350105 x^{8} + 1255282 x^{6} - 17267649 x^{4} - 48409305 x^{2} - 313046839 \)

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:  \(2223111032611629473532068233216=2^{18}\cdot 7^{15}\cdot 97^{3}\cdot 1399^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.52$
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:  $2, 7, 97, 1399$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{7} a^{8}$, $\frac{1}{7} a^{9}$, $\frac{1}{7} a^{10}$, $\frac{1}{7} a^{11}$, $\frac{1}{49} a^{12}$, $\frac{1}{49} a^{13}$, $\frac{1}{4753} a^{14} - \frac{4}{679} a^{12} + \frac{30}{679} a^{10} + \frac{8}{679} a^{8} - \frac{40}{679} a^{6} - \frac{33}{97} a^{4} + \frac{10}{97} a^{2}$, $\frac{1}{4753} a^{15} - \frac{4}{679} a^{13} + \frac{30}{679} a^{11} + \frac{8}{679} a^{9} - \frac{40}{679} a^{7} - \frac{33}{97} a^{5} + \frac{10}{97} a^{3}$, $\frac{1}{1659615270308321886815026037} a^{16} - \frac{81768111052968788648455}{1659615270308321886815026037} a^{14} - \frac{16870110877569309840696347}{1659615270308321886815026037} a^{12} + \frac{11902124795250238811224236}{237087895758331698116432291} a^{10} - \frac{1245791031345570141791986}{237087895758331698116432291} a^{8} - \frac{12963997240197616513738501}{237087895758331698116432291} a^{6} + \frac{3491629927250985135504667}{33869699394047385445204613} a^{4} - \frac{5426250419279435125923}{12040419265569635778601} a^{2} + \frac{959633966355907591939}{3599712976304324098757}$, $\frac{1}{1659615270308321886815026037} a^{17} - \frac{81768111052968788648455}{1659615270308321886815026037} a^{15} - \frac{16870110877569309840696347}{1659615270308321886815026037} a^{13} + \frac{11902124795250238811224236}{237087895758331698116432291} a^{11} - \frac{1245791031345570141791986}{237087895758331698116432291} a^{9} - \frac{12963997240197616513738501}{237087895758331698116432291} a^{7} + \frac{3491629927250985135504667}{33869699394047385445204613} a^{5} - \frac{5426250419279435125923}{12040419265569635778601} a^{3} + \frac{959633966355907591939}{3599712976304324098757} a$

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:  \( 193202004.379 \) (assuming GRH)
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

\(\Q(\zeta_{7})^+\), 6.6.104337856.1, 9.3.164590951.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $18$

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
7Data not computed
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.6.3.1$x^{6} - 194 x^{4} + 9409 x^{2} - 22816825$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
1399Data not computed