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Properties

Label 18.18.1234103070...0000.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{33}\cdot 3^{12}\cdot 5^{9}\cdot 7^{12}$
Root discriminant $60.65$
Ramified primes $2, 3, 5, 7$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_3\times S_3^2$ (as 18T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1000, 0, -19600, -24800, 62440, 99680, -57476, -129080, 9856, 75280, 11144, -21000, -5980, 2520, 1022, -80, -56, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 56*x^16 - 80*x^15 + 1022*x^14 + 2520*x^13 - 5980*x^12 - 21000*x^11 + 11144*x^10 + 75280*x^9 + 9856*x^8 - 129080*x^7 - 57476*x^6 + 99680*x^5 + 62440*x^4 - 24800*x^3 - 19600*x^2 + 1000)
 
gp: K = bnfinit(x^18 - 56*x^16 - 80*x^15 + 1022*x^14 + 2520*x^13 - 5980*x^12 - 21000*x^11 + 11144*x^10 + 75280*x^9 + 9856*x^8 - 129080*x^7 - 57476*x^6 + 99680*x^5 + 62440*x^4 - 24800*x^3 - 19600*x^2 + 1000, 1)
 

Normalized defining polynomial

\( x^{18} - 56 x^{16} - 80 x^{15} + 1022 x^{14} + 2520 x^{13} - 5980 x^{12} - 21000 x^{11} + 11144 x^{10} + 75280 x^{9} + 9856 x^{8} - 129080 x^{7} - 57476 x^{6} + 99680 x^{5} + 62440 x^{4} - 24800 x^{3} - 19600 x^{2} + 1000 \)

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:  \(123410307017276135571456000000000=2^{33}\cdot 3^{12}\cdot 5^{9}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.65$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{4} a^{12}$, $\frac{1}{4} a^{13}$, $\frac{1}{20} a^{14} - \frac{1}{20} a^{12} + \frac{1}{10} a^{10} + \frac{1}{5} a^{6} - \frac{1}{5} a^{4} + \frac{1}{5} a^{2}$, $\frac{1}{260} a^{15} - \frac{3}{260} a^{14} - \frac{21}{260} a^{13} - \frac{1}{130} a^{12} - \frac{9}{130} a^{11} - \frac{4}{65} a^{10} - \frac{3}{13} a^{9} - \frac{3}{26} a^{8} - \frac{14}{65} a^{7} + \frac{29}{130} a^{6} - \frac{6}{65} a^{5} - \frac{27}{65} a^{4} - \frac{24}{65} a^{3} + \frac{7}{65} a^{2} + \frac{1}{13}$, $\frac{1}{1300} a^{16} + \frac{9}{1300} a^{14} - \frac{1}{10} a^{13} + \frac{33}{325} a^{12} + \frac{16}{65} a^{11} + \frac{1}{13} a^{10} + \frac{1}{26} a^{9} + \frac{61}{325} a^{8} + \frac{14}{65} a^{7} - \frac{21}{325} a^{6} - \frac{9}{65} a^{5} + \frac{116}{325} a^{4} - \frac{2}{5} a^{3} + \frac{5}{13} a^{2} - \frac{5}{13} a - \frac{2}{13}$, $\frac{1}{57998854486594695700} a^{17} - \frac{351799533030018}{14499713621648673925} a^{16} - \frac{21865106627513829}{14499713621648673925} a^{15} - \frac{321355233345152017}{14499713621648673925} a^{14} + \frac{3944336935444338817}{57998854486594695700} a^{13} - \frac{2917247681457580747}{28999427243297347850} a^{12} + \frac{455284660073462313}{2899942724329734785} a^{11} + \frac{1158945628812906777}{5799885448659469570} a^{10} + \frac{2194354549362312747}{28999427243297347850} a^{9} + \frac{207787365877940253}{14499713621648673925} a^{8} + \frac{109939769601130431}{2230725172561334450} a^{7} - \frac{2035370500196069521}{28999427243297347850} a^{6} - \frac{6118351460307040744}{14499713621648673925} a^{5} - \frac{5914822310633782567}{14499713621648673925} a^{4} - \frac{1381361357352929548}{2899942724329734785} a^{3} + \frac{500792017791376842}{2899942724329734785} a^{2} - \frac{286180036091029442}{579988544865946957} a + \frac{178585938211022843}{579988544865946957}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 14691002887.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T43):

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

\(\Q(\sqrt{10}) \), \(\Q(\zeta_{7})^+\), 6.6.153664000.1, 6.6.49787136000.1

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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.9.12.16$x^{9} + 9 x^{5} + 18 x^{3} + 27 x^{2} + 27$$3$$3$$12$$S_3\times C_3$$[2]^{6}$
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7Data not computed