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Properties

Label 18.18.2441094094...0784.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 7^{12}\cdot 53^{6}\cdot 181^{5}$
Root discriminant $92.46$
Ramified primes $2, 7, 53, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T176

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-219172337, -907486600, 7204566626, 2788171094, -6161937706, -2126342036, 1635451266, 478686318, -211154628, -48293558, 15046066, 2539072, -621949, -72108, 14860, 1044, -190, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 190*x^16 + 1044*x^15 + 14860*x^14 - 72108*x^13 - 621949*x^12 + 2539072*x^11 + 15046066*x^10 - 48293558*x^9 - 211154628*x^8 + 478686318*x^7 + 1635451266*x^6 - 2126342036*x^5 - 6161937706*x^4 + 2788171094*x^3 + 7204566626*x^2 - 907486600*x - 219172337)
 
gp: K = bnfinit(x^18 - 6*x^17 - 190*x^16 + 1044*x^15 + 14860*x^14 - 72108*x^13 - 621949*x^12 + 2539072*x^11 + 15046066*x^10 - 48293558*x^9 - 211154628*x^8 + 478686318*x^7 + 1635451266*x^6 - 2126342036*x^5 - 6161937706*x^4 + 2788171094*x^3 + 7204566626*x^2 - 907486600*x - 219172337, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 190 x^{16} + 1044 x^{15} + 14860 x^{14} - 72108 x^{13} - 621949 x^{12} + 2539072 x^{11} + 15046066 x^{10} - 48293558 x^{9} - 211154628 x^{8} + 478686318 x^{7} + 1635451266 x^{6} - 2126342036 x^{5} - 6161937706 x^{4} + 2788171094 x^{3} + 7204566626 x^{2} - 907486600 x - 219172337 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(244109409460733537552375387021430784=2^{12}\cdot 7^{12}\cdot 53^{6}\cdot 181^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.46$
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:  $2, 7, 53, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8713328056665566249578019621500805896710085796582254300606222140521512538066} a^{17} + \frac{1711320877670305974267384708167585607007108781386330514012103364234938164645}{8713328056665566249578019621500805896710085796582254300606222140521512538066} a^{16} - \frac{1060244387016464103326590265677302093000964987177429138183030188543916260640}{4356664028332783124789009810750402948355042898291127150303111070260756269033} a^{15} - \frac{1616174412810228745747892369189971942667694476476865795619435436487422890276}{4356664028332783124789009810750402948355042898291127150303111070260756269033} a^{14} + \frac{986620066279262835946626856297592394973241734066887164553606696613538057985}{8713328056665566249578019621500805896710085796582254300606222140521512538066} a^{13} + \frac{1644066996228334990113412375358106639096529403087190681172914262339593898508}{4356664028332783124789009810750402948355042898291127150303111070260756269033} a^{12} + \frac{840277197685466683385338245851802546716095324317154390359119485169489341054}{4356664028332783124789009810750402948355042898291127150303111070260756269033} a^{11} - \frac{340781555909089431239659510816108626159633315241269486069670921694947703838}{4356664028332783124789009810750402948355042898291127150303111070260756269033} a^{10} + \frac{4130117826088279839321882916374096426221580285889470985986585355244719639635}{8713328056665566249578019621500805896710085796582254300606222140521512538066} a^{9} + \frac{2019648886361656124269130447374820322715415730795849608157782516571947481382}{4356664028332783124789009810750402948355042898291127150303111070260756269033} a^{8} + \frac{2389336373211159172248667327330717972161453598118004296751440723578795317501}{8713328056665566249578019621500805896710085796582254300606222140521512538066} a^{7} + \frac{1558794329872120601501880480875154261634736297269235638413091228561007261693}{8713328056665566249578019621500805896710085796582254300606222140521512538066} a^{6} - \frac{975332897865765560334860547128651803846084365686652606263975071582140682379}{8713328056665566249578019621500805896710085796582254300606222140521512538066} a^{5} - \frac{1863449089567291317136693426732364093740787721948714347852864957101151689}{6904380393554331418049143915610781217678356415675320364981158589953654943} a^{4} + \frac{1117870216613191518952769215658281878248322309542596416138588704805926154266}{4356664028332783124789009810750402948355042898291127150303111070260756269033} a^{3} - \frac{2047543571639453084118153692603257035062916796793211132163736421690990301025}{8713328056665566249578019621500805896710085796582254300606222140521512538066} a^{2} - \frac{2017055055623110000089357373730151632045011498791391164500483559206499421053}{4356664028332783124789009810750402948355042898291127150303111070260756269033} a + \frac{1800321218385221739848813469045253309585355034821172010399437216301156506437}{8713328056665566249578019621500805896710085796582254300606222140521512538066}$

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:  $17$
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:  \( 676604288950 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T176:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 40 conjugacy class representatives for t18n176
Character table for t18n176 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.2597.1, 9.9.17515230173.1

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 R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
7Data not computed
$53$53.6.0.1$x^{6} - x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
53.12.6.1$x^{12} + 2382032 x^{6} - 418195493 x^{2} + 1418519112256$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
181Data not computed