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Properties

Label 18.0.17224401851...9168.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 271^{10}$
Root discriminant $61.79$
Ramified primes $2, 3, 271$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![531441, 2243862, 3332988, 2226366, 857304, 227691, 164835, 39366, 20628, 27540, 4617, -306, 1167, -612, -122, 51, -5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 5*x^16 + 51*x^15 - 122*x^14 - 612*x^13 + 1167*x^12 - 306*x^11 + 4617*x^10 + 27540*x^9 + 20628*x^8 + 39366*x^7 + 164835*x^6 + 227691*x^5 + 857304*x^4 + 2226366*x^3 + 3332988*x^2 + 2243862*x + 531441)
 
gp: K = bnfinit(x^18 - 3*x^17 - 5*x^16 + 51*x^15 - 122*x^14 - 612*x^13 + 1167*x^12 - 306*x^11 + 4617*x^10 + 27540*x^9 + 20628*x^8 + 39366*x^7 + 164835*x^6 + 227691*x^5 + 857304*x^4 + 2226366*x^3 + 3332988*x^2 + 2243862*x + 531441, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 5 x^{16} + 51 x^{15} - 122 x^{14} - 612 x^{13} + 1167 x^{12} - 306 x^{11} + 4617 x^{10} + 27540 x^{9} + 20628 x^{8} + 39366 x^{7} + 164835 x^{6} + 227691 x^{5} + 857304 x^{4} + 2226366 x^{3} + 3332988 x^{2} + 2243862 x + 531441 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-172244018518353824873264845959168=-\,2^{12}\cdot 3^{9}\cdot 271^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.79$
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:  $2, 3, 271$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{2}{9} a^{4} - \frac{1}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} - \frac{1}{3} a^{4} - \frac{2}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{5} - \frac{1}{9} a^{4} + \frac{4}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{9} + \frac{1}{27} a^{7} + \frac{1}{27} a^{5} + \frac{1}{9} a^{4} - \frac{2}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{81} a^{10} + \frac{1}{81} a^{8} + \frac{1}{81} a^{6} + \frac{1}{27} a^{5} + \frac{7}{27} a^{4} + \frac{1}{3} a^{3} + \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{81} a^{11} + \frac{1}{81} a^{9} + \frac{1}{81} a^{7} + \frac{1}{27} a^{6} - \frac{2}{27} a^{5} + \frac{4}{9} a^{3}$, $\frac{1}{243} a^{12} + \frac{1}{243} a^{10} + \frac{10}{243} a^{8} + \frac{4}{81} a^{7} + \frac{4}{81} a^{6} - \frac{1}{27} a^{5} - \frac{8}{27} a^{4} + \frac{4}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{729} a^{13} - \frac{2}{729} a^{11} + \frac{7}{729} a^{9} - \frac{5}{243} a^{8} - \frac{2}{81} a^{7} + \frac{1}{81} a^{6} - \frac{1}{27} a^{5} - \frac{8}{27} a^{4} - \frac{13}{27} a^{3} + \frac{1}{3} a$, $\frac{1}{2187} a^{14} - \frac{2}{2187} a^{12} + \frac{7}{2187} a^{10} - \frac{5}{729} a^{9} - \frac{11}{243} a^{8} + \frac{1}{243} a^{7} - \frac{4}{81} a^{6} + \frac{1}{81} a^{5} - \frac{7}{81} a^{4} + \frac{1}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{6561} a^{15} + \frac{4}{6561} a^{13} + \frac{1}{729} a^{12} - \frac{5}{6561} a^{11} - \frac{11}{2187} a^{10} - \frac{19}{2187} a^{9} - \frac{29}{729} a^{8} - \frac{4}{243} a^{7} - \frac{4}{81} a^{6} + \frac{26}{243} a^{5} + \frac{8}{81} a^{4} + \frac{13}{81} a^{3} - \frac{10}{27} a^{2} + \frac{4}{9} a$, $\frac{1}{59049} a^{16} + \frac{1}{19683} a^{15} + \frac{13}{59049} a^{14} - \frac{11}{19683} a^{13} - \frac{77}{59049} a^{12} + \frac{74}{19683} a^{11} - \frac{58}{19683} a^{10} - \frac{2}{2187} a^{9} + \frac{23}{729} a^{8} + \frac{40}{729} a^{7} - \frac{1}{2187} a^{6} + \frac{79}{729} a^{5} - \frac{245}{729} a^{4} - \frac{34}{81} a^{3} - \frac{2}{9} a^{2} - \frac{11}{27} a + \frac{1}{3}$, $\frac{1}{17565549019864566655481791841781} a^{17} + \frac{43721706204505341627962957}{5855183006621522218493930613927} a^{16} - \frac{1273673112648538122007945898}{17565549019864566655481791841781} a^{15} + \frac{1033029008320724556232740875}{5855183006621522218493930613927} a^{14} - \frac{8386778030984172465380885813}{17565549019864566655481791841781} a^{13} - \frac{119448758708330256786050153}{1951727668873840739497976871309} a^{12} + \frac{5369448204933099661785735704}{5855183006621522218493930613927} a^{11} - \frac{7057431671170668369510170635}{1951727668873840739497976871309} a^{10} - \frac{2311600002376285334858775446}{216858629874871193277552985701} a^{9} - \frac{5191173062518290245619307265}{216858629874871193277552985701} a^{8} - \frac{28632495143484233362879146991}{650575889624613579832658957103} a^{7} + \frac{1330410324530119064341603357}{24095403319430132586394776189} a^{6} + \frac{12957450871334682426362214976}{216858629874871193277552985701} a^{5} + \frac{32799425196992805211004626474}{72286209958290397759184328567} a^{4} - \frac{872756003955744106513440896}{8031801106476710862131592063} a^{3} - \frac{903990060546224773052179556}{8031801106476710862131592063} a^{2} + \frac{1014902136031544130587900593}{2677267035492236954043864021} a - \frac{61629099503231774632903823}{297474115054692994893762669}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{24801728435049142}{1742762093550236266401} a^{17} + \frac{29562582292277314}{580920697850078755467} a^{16} + \frac{73856916550353128}{1742762093550236266401} a^{15} - \frac{437173978811749313}{580920697850078755467} a^{14} + \frac{3781623310206322937}{1742762093550236266401} a^{13} + \frac{483507781363455196}{64546744205564306163} a^{12} - \frac{12227542213016016047}{580920697850078755467} a^{11} + \frac{3159880428648480070}{193640232616692918489} a^{10} - \frac{1590770936861051230}{21515581401854768721} a^{9} - \frac{7569098207153650537}{21515581401854768721} a^{8} - \frac{5615437335211633724}{64546744205564306163} a^{7} - \frac{3505108359472258225}{7171860467284922907} a^{6} - \frac{45008829005758515298}{21515581401854768721} a^{5} - \frac{14395281397763747182}{7171860467284922907} a^{4} - \frac{8762134816891953868}{796873385253880323} a^{3} - \frac{20308919815067173471}{796873385253880323} a^{2} - \frac{8602048548004184641}{265624461751293441} a - \frac{352945352737255009}{29513829083477049} \) (order $6$)
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:  \( 3453478086.089924 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.3252.1, 6.0.1982907.1, 6.0.31726512.1

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

Sibling fields

Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
271Data not computed