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Properties

Label 18.8.19597718879...0608.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,2^{12}\cdot 3^{30}\cdot 7^{6}\cdot 12547^{3}$
Root discriminant $91.34$
Ramified primes $2, 3, 7, 12547$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T366

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3707657, 33278943, 63884925, 35668848, 2811813, -33240231, -32868075, -20503479, -7474338, -1174006, 230385, 207387, 53031, 5403, -873, -249, -63, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 63*x^16 - 249*x^15 - 873*x^14 + 5403*x^13 + 53031*x^12 + 207387*x^11 + 230385*x^10 - 1174006*x^9 - 7474338*x^8 - 20503479*x^7 - 32868075*x^6 - 33240231*x^5 + 2811813*x^4 + 35668848*x^3 + 63884925*x^2 + 33278943*x + 3707657)
 
gp: K = bnfinit(x^18 - 63*x^16 - 249*x^15 - 873*x^14 + 5403*x^13 + 53031*x^12 + 207387*x^11 + 230385*x^10 - 1174006*x^9 - 7474338*x^8 - 20503479*x^7 - 32868075*x^6 - 33240231*x^5 + 2811813*x^4 + 35668848*x^3 + 63884925*x^2 + 33278943*x + 3707657, 1)
 

Normalized defining polynomial

\( x^{18} - 63 x^{16} - 249 x^{15} - 873 x^{14} + 5403 x^{13} + 53031 x^{12} + 207387 x^{11} + 230385 x^{10} - 1174006 x^{9} - 7474338 x^{8} - 20503479 x^{7} - 32868075 x^{6} - 33240231 x^{5} + 2811813 x^{4} + 35668848 x^{3} + 63884925 x^{2} + 33278943 x + 3707657 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-195977188797288322153846057926340608=-\,2^{12}\cdot 3^{30}\cdot 7^{6}\cdot 12547^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.34$
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:  $2, 3, 7, 12547$
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}{156949857571780035852601582553348845667100675110189686360993840612226343299} a^{17} - \frac{59507104286251477370213613568255313582175188155038917724277804912801095991}{156949857571780035852601582553348845667100675110189686360993840612226343299} a^{16} - \frac{62472379844403444402197106128044168785871833582637313983076042680477669004}{156949857571780035852601582553348845667100675110189686360993840612226343299} a^{15} + \frac{62319321055518715098946242251602766701398130824530492279594794805373898238}{156949857571780035852601582553348845667100675110189686360993840612226343299} a^{14} + \frac{25951708796097508587190505477926285598508599554196121570246063515608102292}{156949857571780035852601582553348845667100675110189686360993840612226343299} a^{13} + \frac{65001721779137157925390510758767184004238147615271759882421208973271914393}{156949857571780035852601582553348845667100675110189686360993840612226343299} a^{12} - \frac{59836774333715156982133702972776791513704638933690633665427937123756020591}{156949857571780035852601582553348845667100675110189686360993840612226343299} a^{11} + \frac{40764338908897699883359430164003734862839222139684415388722758602934536374}{156949857571780035852601582553348845667100675110189686360993840612226343299} a^{10} + \frac{36485519038206993175279702077882122955565331806347320565175442832232508762}{156949857571780035852601582553348845667100675110189686360993840612226343299} a^{9} - \frac{56896151515747654671523659572120219737999179040197900767400196963353955654}{156949857571780035852601582553348845667100675110189686360993840612226343299} a^{8} + \frac{39960943613069611446213379674161965957345984891475556550623176893963419599}{156949857571780035852601582553348845667100675110189686360993840612226343299} a^{7} - \frac{56902351650693402210656463619505127986183928925151797578159503861197526072}{156949857571780035852601582553348845667100675110189686360993840612226343299} a^{6} + \frac{63268710847647341705624525877721104506632614913483859904090374050627824119}{156949857571780035852601582553348845667100675110189686360993840612226343299} a^{5} + \frac{58101210767515129901470574881739000914237064412275630475364053389047388989}{156949857571780035852601582553348845667100675110189686360993840612226343299} a^{4} - \frac{26867166085457997295433827974081249978453833809659781974591350813491827987}{156949857571780035852601582553348845667100675110189686360993840612226343299} a^{3} + \frac{54140594171954496458770958573381157927042360569697754268866385957554246973}{156949857571780035852601582553348845667100675110189686360993840612226343299} a^{2} + \frac{64592631081694070810374446647460260234724497873754038014095023213268425475}{156949857571780035852601582553348845667100675110189686360993840612226343299} a + \frac{5610469896767334318971210577868355986115492580968476748930843348327173963}{156949857571780035852601582553348845667100675110189686360993840612226343299}$

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:  $12$
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:  \( 59209132152.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T366:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 48 conjugacy class representatives for t18n366
Character table for t18n366 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.756.1, 9.9.314987206464.1

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
3Data not computed
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
12547Data not computed