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Properties

Label 18.2.24724476901...3873.3
Degree $18$
Signature $[2, 8]$
Discriminant $7^{12}\cdot 97^{3}\cdot 1399^{2}$
Root discriminant $17.54$
Ramified primes $7, 97, 1399$
Class number $1$
Class group Trivial
Galois group 18T879

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-139, 223, -155, 250, -169, 24, -32, -138, 196, -143, 172, -137, 93, -63, 29, -18, 11, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 11*x^16 - 18*x^15 + 29*x^14 - 63*x^13 + 93*x^12 - 137*x^11 + 172*x^10 - 143*x^9 + 196*x^8 - 138*x^7 - 32*x^6 + 24*x^5 - 169*x^4 + 250*x^3 - 155*x^2 + 223*x - 139)
 
gp: K = bnfinit(x^18 - 4*x^17 + 11*x^16 - 18*x^15 + 29*x^14 - 63*x^13 + 93*x^12 - 137*x^11 + 172*x^10 - 143*x^9 + 196*x^8 - 138*x^7 - 32*x^6 + 24*x^5 - 169*x^4 + 250*x^3 - 155*x^2 + 223*x - 139, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 11 x^{16} - 18 x^{15} + 29 x^{14} - 63 x^{13} + 93 x^{12} - 137 x^{11} + 172 x^{10} - 143 x^{9} + 196 x^{8} - 138 x^{7} - 32 x^{6} + 24 x^{5} - 169 x^{4} + 250 x^{3} - 155 x^{2} + 223 x - 139 \)

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:  \(24724476901703653513873=7^{12}\cdot 97^{3}\cdot 1399^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.54$
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:  $7, 97, 1399$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{11} a^{16} + \frac{2}{11} a^{15} + \frac{3}{11} a^{14} + \frac{4}{11} a^{13} + \frac{4}{11} a^{12} + \frac{2}{11} a^{11} + \frac{3}{11} a^{10} - \frac{5}{11} a^{9} + \frac{5}{11} a^{8} - \frac{2}{11} a^{7} - \frac{4}{11} a^{6} - \frac{1}{11} a^{5} - \frac{2}{11} a^{4} - \frac{1}{11} a^{3} - \frac{3}{11} a^{2} - \frac{1}{11} a - \frac{2}{11}$, $\frac{1}{1620275958835963} a^{17} - \frac{6099913740361}{1620275958835963} a^{16} - \frac{131136242430086}{1620275958835963} a^{15} - \frac{608363120911419}{1620275958835963} a^{14} - \frac{379002569627096}{1620275958835963} a^{13} - \frac{145257841454186}{1620275958835963} a^{12} - \frac{806395909310500}{1620275958835963} a^{11} - \frac{642943929261131}{1620275958835963} a^{10} + \frac{384283283778689}{1620275958835963} a^{9} + \frac{509476529001940}{1620275958835963} a^{8} - \frac{95430793619104}{1620275958835963} a^{7} - \frac{57978210116452}{147297814439633} a^{6} - \frac{280442521380306}{1620275958835963} a^{5} + \frac{10671649829595}{55871584787447} a^{4} + \frac{20378129974}{13390710403603} a^{3} + \frac{47781651747284}{124636612218151} a^{2} - \frac{170583907382173}{1620275958835963} a - \frac{315715127871357}{1620275958835963}$

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:  \( 7058.90122134 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T879:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 331776
The 360 conjugacy class representatives for t18n879 are not computed
Character table for t18n879 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ $18$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $18$

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$$0$Trivial$[\ ]$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.2$x^{4} - 97 x^{2} + 47045$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
97.6.0.1$x^{6} - x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$
1399Data not computed