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Properties

Label 18.0.25014102650...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{18}\cdot 5^{8}\cdot 7^{9}$
Root discriminant $25.76$
Ramified primes $2, 3, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_3^2:S_3$ (as 18T52)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![512, -1536, 5760, -6592, 9120, -3456, 2800, -804, 894, -635, 456, -129, 23, 9, 0, -8, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 9*x^16 - 8*x^15 + 9*x^13 + 23*x^12 - 129*x^11 + 456*x^10 - 635*x^9 + 894*x^8 - 804*x^7 + 2800*x^6 - 3456*x^5 + 9120*x^4 - 6592*x^3 + 5760*x^2 - 1536*x + 512)
 
gp: K = bnfinit(x^18 - 3*x^17 + 9*x^16 - 8*x^15 + 9*x^13 + 23*x^12 - 129*x^11 + 456*x^10 - 635*x^9 + 894*x^8 - 804*x^7 + 2800*x^6 - 3456*x^5 + 9120*x^4 - 6592*x^3 + 5760*x^2 - 1536*x + 512, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 9 x^{16} - 8 x^{15} + 9 x^{13} + 23 x^{12} - 129 x^{11} + 456 x^{10} - 635 x^{9} + 894 x^{8} - 804 x^{7} + 2800 x^{6} - 3456 x^{5} + 9120 x^{4} - 6592 x^{3} + 5760 x^{2} - 1536 x + 512 \)

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:  \(-25014102650966116800000000=-\,2^{12}\cdot 3^{18}\cdot 5^{8}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.76$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} + \frac{1}{4} a^{9} + \frac{1}{16} a^{8} - \frac{7}{16} a^{7} - \frac{7}{16} a^{6} + \frac{1}{8} a^{5} + \frac{1}{16} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{16} a^{11} + \frac{1}{8} a^{10} + \frac{1}{32} a^{9} - \frac{7}{32} a^{8} + \frac{9}{32} a^{7} - \frac{7}{16} a^{6} + \frac{1}{32} a^{5} - \frac{1}{2} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{320} a^{15} + \frac{3}{320} a^{14} - \frac{1}{64} a^{13} - \frac{3}{160} a^{12} + \frac{7}{80} a^{11} + \frac{49}{320} a^{10} - \frac{67}{320} a^{9} + \frac{9}{64} a^{8} - \frac{69}{160} a^{7} - \frac{119}{320} a^{6} + \frac{29}{80} a^{5} + \frac{29}{80} a^{4} + \frac{9}{40} a^{3} - \frac{3}{20} a^{2} + \frac{3}{10} a - \frac{1}{5}$, $\frac{1}{30080} a^{16} + \frac{1}{6016} a^{15} - \frac{79}{30080} a^{14} + \frac{7}{940} a^{13} + \frac{43}{940} a^{12} - \frac{523}{6016} a^{11} - \frac{3009}{30080} a^{10} - \frac{809}{30080} a^{9} + \frac{51}{940} a^{8} + \frac{33}{6016} a^{7} + \frac{1939}{15040} a^{6} - \frac{2613}{7520} a^{5} - \frac{731}{1880} a^{4} - \frac{211}{470} a^{3} - \frac{22}{47} a^{2} - \frac{2}{5} a - \frac{116}{235}$, $\frac{1}{81161170291798200958720} a^{17} - \frac{299300619533060529}{81161170291798200958720} a^{16} - \frac{39138126655391704817}{81161170291798200958720} a^{15} - \frac{441190510121988793177}{40580585145899100479360} a^{14} + \frac{51279043281013104997}{4058058514589910047936} a^{13} + \frac{2054657771497691814049}{81161170291798200958720} a^{12} - \frac{4040899475023045259703}{81161170291798200958720} a^{11} + \frac{700099396116523884901}{16232234058359640191744} a^{10} + \frac{13788989525367076268577}{40580585145899100479360} a^{9} + \frac{1524410632412397798337}{81161170291798200958720} a^{8} - \frac{163264369076509531861}{634071642904673444990} a^{7} - \frac{1441881242138568414443}{20290292572949550239680} a^{6} + \frac{283689479724875950823}{10145146286474775119840} a^{5} - \frac{1678390713519445257253}{5072573143237387559920} a^{4} - \frac{202046578194358920673}{507257314323738755992} a^{3} + \frac{400763707787630545473}{1268143285809346889980} a^{2} + \frac{73838361682530570792}{317035821452336722495} a + \frac{150939408613097299864}{317035821452336722495}$

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:  \( -1 \) (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:  \( 148711.30288280427 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2:S_3$ (as 18T52):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times C_3^2:S_3$
Character table for $C_2\times C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.1.108.1, 6.0.4000752.4, 9.3.787320000.1

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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$5$5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$