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Properties

Label 18.6.91787645575...6913.1
Degree $18$
Signature $[6, 6]$
Discriminant $97^{9}\cdot 367^{2}\cdot 299401^{2}$
Root discriminant $77.06$
Ramified primes $97, 367, 299401$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T913

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2611950958249, -168842921041, 1008259194697, 51244516941, -171026989792, -6851856807, 16789448405, 528345520, -1053188855, -25765391, 43827399, 815812, -1210662, -16412, 21412, 192, -220, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 220*x^16 + 192*x^15 + 21412*x^14 - 16412*x^13 - 1210662*x^12 + 815812*x^11 + 43827399*x^10 - 25765391*x^9 - 1053188855*x^8 + 528345520*x^7 + 16789448405*x^6 - 6851856807*x^5 - 171026989792*x^4 + 51244516941*x^3 + 1008259194697*x^2 - 168842921041*x - 2611950958249)
 
gp: K = bnfinit(x^18 - x^17 - 220*x^16 + 192*x^15 + 21412*x^14 - 16412*x^13 - 1210662*x^12 + 815812*x^11 + 43827399*x^10 - 25765391*x^9 - 1053188855*x^8 + 528345520*x^7 + 16789448405*x^6 - 6851856807*x^5 - 171026989792*x^4 + 51244516941*x^3 + 1008259194697*x^2 - 168842921041*x - 2611950958249, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 220 x^{16} + 192 x^{15} + 21412 x^{14} - 16412 x^{13} - 1210662 x^{12} + 815812 x^{11} + 43827399 x^{10} - 25765391 x^{9} - 1053188855 x^{8} + 528345520 x^{7} + 16789448405 x^{6} - 6851856807 x^{5} - 171026989792 x^{4} + 51244516941 x^{3} + 1008259194697 x^{2} - 168842921041 x - 2611950958249 \)

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:  \(9178764557539239451556240651976913=97^{9}\cdot 367^{2}\cdot 299401^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.06$
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:  $97, 367, 299401$
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}{270078039697308464981228536899815229262073943268647238333053461923220369} a^{17} - \frac{96759375334464282369056076383703258653748268276457205642456359672741097}{270078039697308464981228536899815229262073943268647238333053461923220369} a^{16} - \frac{129159592076405983280729534183355677244054715088917184473922740513456038}{270078039697308464981228536899815229262073943268647238333053461923220369} a^{15} - \frac{63116604027208660426114171187861585920554034398033400553515738975872079}{270078039697308464981228536899815229262073943268647238333053461923220369} a^{14} - \frac{15366585127545013414964937176848204494525554758166912170224848272260318}{270078039697308464981228536899815229262073943268647238333053461923220369} a^{13} + \frac{44879878935331462922043223422182182895216215237299421642625648629761860}{270078039697308464981228536899815229262073943268647238333053461923220369} a^{12} + \frac{6992133598906330347849212520203290021532976598005853939257716186154565}{15886943511606380293013443347047954662474937839332190490179615407248257} a^{11} - \frac{106621468359331970968425885641608746975414537716689986853540280470237491}{270078039697308464981228536899815229262073943268647238333053461923220369} a^{10} - \frac{45438711206216339687254012341760085505496744176588851775423042057017883}{270078039697308464981228536899815229262073943268647238333053461923220369} a^{9} + \frac{59252538001709382893731721661216624914303083867040521355580586972378280}{270078039697308464981228536899815229262073943268647238333053461923220369} a^{8} + \frac{103581510035415255103770694159861283454426999283691915624331507695147727}{270078039697308464981228536899815229262073943268647238333053461923220369} a^{7} - \frac{21158960428574548853030164825949713663656881022416043923812319443339217}{270078039697308464981228536899815229262073943268647238333053461923220369} a^{6} + \frac{85678661413370689057002895509622493855708224182696504547065196147070819}{270078039697308464981228536899815229262073943268647238333053461923220369} a^{5} + \frac{6968036370090482789217758562811683672744548328544444465782735514711197}{15886943511606380293013443347047954662474937839332190490179615407248257} a^{4} + \frac{87074778020351916851874161154147095193771551357577451431349748061587777}{270078039697308464981228536899815229262073943268647238333053461923220369} a^{3} + \frac{111548924122958243150279888695874876576254056196053970601107454443225971}{270078039697308464981228536899815229262073943268647238333053461923220369} a^{2} - \frac{364183530734718334809034335744961883627169586843520598894027760570032}{270078039697308464981228536899815229262073943268647238333053461923220369} a - \frac{129528925423727225123140543047349989599241117483389138569228554704667150}{270078039697308464981228536899815229262073943268647238333053461923220369}$

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:  \( 9055457367.61 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T913:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 725760
The 60 conjugacy class representatives for t18n913 are not computed
Character table for t18n913 is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 9.3.109880167.1

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

Sibling fields

Degree 18 sibling: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ $18$ ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$97$97.4.2.1$x^{4} + 873 x^{2} + 235225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
97.6.3.1$x^{6} - 194 x^{4} + 9409 x^{2} - 22816825$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
97.8.4.1$x^{8} + 432814 x^{4} - 912673 x^{2} + 46831989649$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
367Data not computed
299401Data not computed