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Properties

Label 18.2.64973982181...0193.1
Degree $18$
Signature $[2, 8]$
Discriminant $7^{12}\cdot 97^{3}\cdot 22679^{2}$
Root discriminant $23.90$
Ramified primes $7, 97, 22679$
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, 0, 0, 10, -84, 371, -754, 1330, -1225, 1146, -581, 364, -99, 42, 21, -1, 7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 7*x^16 - x^15 + 21*x^14 + 42*x^13 - 99*x^12 + 364*x^11 - 581*x^10 + 1146*x^9 - 1225*x^8 + 1330*x^7 - 754*x^6 + 371*x^5 - 84*x^4 + 10*x^3 - 1)
 
gp: K = bnfinit(x^18 + 7*x^16 - x^15 + 21*x^14 + 42*x^13 - 99*x^12 + 364*x^11 - 581*x^10 + 1146*x^9 - 1225*x^8 + 1330*x^7 - 754*x^6 + 371*x^5 - 84*x^4 + 10*x^3 - 1, 1)
 

Normalized defining polynomial

\( x^{18} + 7 x^{16} - x^{15} + 21 x^{14} + 42 x^{13} - 99 x^{12} + 364 x^{11} - 581 x^{10} + 1146 x^{9} - 1225 x^{8} + 1330 x^{7} - 754 x^{6} + 371 x^{5} - 84 x^{4} + 10 x^{3} - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6497398218116128597530193=7^{12}\cdot 97^{3}\cdot 22679^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.90$
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:  $7, 97, 22679$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{116697194715251116471} a^{17} - \frac{45501005965333971832}{116697194715251116471} a^{16} - \frac{14395358122477166890}{116697194715251116471} a^{15} + \frac{21337702222943393889}{116697194715251116471} a^{14} - \frac{502430224531028561}{1203063863043825943} a^{13} - \frac{14396905624193923350}{116697194715251116471} a^{12} - \frac{34405196426766061414}{116697194715251116471} a^{11} + \frac{8846861778482352098}{116697194715251116471} a^{10} + \frac{21627098408244702701}{116697194715251116471} a^{9} + \frac{19762002518482688303}{116697194715251116471} a^{8} + \frac{15828203943027442718}{116697194715251116471} a^{7} - \frac{20051858359197497593}{116697194715251116471} a^{6} - \frac{32180409159399923322}{116697194715251116471} a^{5} - \frac{9167506061109440666}{116697194715251116471} a^{4} + \frac{35064234707881988446}{116697194715251116471} a^{3} - \frac{6829583333747460222}{116697194715251116471} a^{2} + \frac{31171375583013579567}{116697194715251116471} a - \frac{13669282104635466443}{116697194715251116471}$

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:  $9$
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:  \( 88562.6720172 \)
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

\(\Q(\zeta_{7})^+\), 6.2.232897.1, 9.7.2668161671.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $18$

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
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}$
97.6.3.2$x^{6} - 9409 x^{2} + 4563365$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
22679Data not computed