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Properties

Label 18.6.13770649837...4057.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 41^{5}\cdot 97^{5}$
Root discriminant $36.58$
Ramified primes $7, 41, 97$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T401

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-175616, -175616, 175616, 159936, 42336, -63504, -81088, 8652, 32550, 4271, -6494, -2168, 547, 333, 57, -12, -19, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 19*x^16 - 12*x^15 + 57*x^14 + 333*x^13 + 547*x^12 - 2168*x^11 - 6494*x^10 + 4271*x^9 + 32550*x^8 + 8652*x^7 - 81088*x^6 - 63504*x^5 + 42336*x^4 + 159936*x^3 + 175616*x^2 - 175616*x - 175616)
 
gp: K = bnfinit(x^18 - x^17 - 19*x^16 - 12*x^15 + 57*x^14 + 333*x^13 + 547*x^12 - 2168*x^11 - 6494*x^10 + 4271*x^9 + 32550*x^8 + 8652*x^7 - 81088*x^6 - 63504*x^5 + 42336*x^4 + 159936*x^3 + 175616*x^2 - 175616*x - 175616, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 19 x^{16} - 12 x^{15} + 57 x^{14} + 333 x^{13} + 547 x^{12} - 2168 x^{11} - 6494 x^{10} + 4271 x^{9} + 32550 x^{8} + 8652 x^{7} - 81088 x^{6} - 63504 x^{5} + 42336 x^{4} + 159936 x^{3} + 175616 x^{2} - 175616 x - 175616 \)

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:  \(13770649837088816389593674057=7^{12}\cdot 41^{5}\cdot 97^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.58$
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:  $7, 41, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{56} a^{12} - \frac{1}{56} a^{11} + \frac{9}{56} a^{10} + \frac{2}{7} a^{9} - \frac{27}{56} a^{8} - \frac{3}{56} a^{7} + \frac{15}{56} a^{6} - \frac{3}{14} a^{5} - \frac{13}{28} a^{4} + \frac{15}{56} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{112} a^{13} - \frac{1}{112} a^{12} + \frac{9}{112} a^{11} + \frac{1}{7} a^{10} + \frac{29}{112} a^{9} + \frac{53}{112} a^{8} + \frac{15}{112} a^{7} + \frac{11}{28} a^{6} + \frac{15}{56} a^{5} - \frac{41}{112} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{224} a^{14} - \frac{1}{224} a^{13} + \frac{1}{224} a^{12} + \frac{3}{28} a^{11} - \frac{43}{224} a^{10} + \frac{37}{224} a^{9} + \frac{1}{32} a^{8} + \frac{17}{56} a^{7} - \frac{45}{112} a^{6} - \frac{57}{224} a^{5} - \frac{1}{112} a^{4} + \frac{13}{28} a^{3}$, $\frac{1}{9408} a^{15} - \frac{1}{9408} a^{14} + \frac{3}{3136} a^{13} + \frac{1}{588} a^{12} + \frac{253}{9408} a^{11} + \frac{1285}{9408} a^{10} - \frac{881}{9408} a^{9} + \frac{235}{2352} a^{8} + \frac{1583}{4704} a^{7} + \frac{71}{9408} a^{6} + \frac{107}{224} a^{5} - \frac{1}{42} a^{4} + \frac{5}{28} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{1825152} a^{16} - \frac{37}{1825152} a^{15} + \frac{71}{608384} a^{14} + \frac{43}{65184} a^{13} + \frac{14965}{1825152} a^{12} - \frac{193631}{1825152} a^{11} + \frac{39043}{1825152} a^{10} + \frac{69245}{228144} a^{9} - \frac{8053}{18624} a^{8} - \frac{541633}{1825152} a^{7} - \frac{35685}{304192} a^{6} - \frac{2533}{8148} a^{5} + \frac{73}{776} a^{4} - \frac{1511}{4074} a^{3} - \frac{149}{1164} a^{2} - \frac{107}{582} a - \frac{14}{97}$, $\frac{1}{1527601720787568144265303416576} a^{17} + \frac{373850794377859695610781}{1527601720787568144265303416576} a^{16} - \frac{10567695529979493480691191}{509200573595856048088434472192} a^{15} + \frac{485639885362814898740264819}{763800860393784072132651708288} a^{14} + \frac{6303129002608322126979882181}{1527601720787568144265303416576} a^{13} - \frac{7072294627386704933155064141}{1527601720787568144265303416576} a^{12} - \frac{41764279989974160080934941243}{1527601720787568144265303416576} a^{11} + \frac{3052483600345659211285928315}{763800860393784072132651708288} a^{10} + \frac{7200632473130315469802895513}{109114408627683438876093101184} a^{9} + \frac{296156396738171060471816074115}{1527601720787568144265303416576} a^{8} + \frac{7793340680425061773092954129}{31825035849741003005527154512} a^{7} - \frac{9635396384702940976794176279}{381900430196892036066325854144} a^{6} + \frac{2058484947925878383949072975}{9092867385640286573007758432} a^{5} + \frac{771323700111801861585160207}{3409825269615107464877909412} a^{4} - \frac{143395984263557759823638509}{487117895659301066411129916} a^{3} - \frac{175455202878012597139407143}{487117895659301066411129916} a^{2} + \frac{26736857798543559403434757}{81186315943216844401854986} a - \frac{10085202716260502151442550}{40593157971608422200927493}$

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:  \( 10779495.3631 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T401:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2592
The 44 conjugacy class representatives for t18n401
Character table for t18n401 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.2.9548777.2, 9.5.467890073.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
41Data 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}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$