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Properties

Label 18.0.10564318602...5856.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{20}\cdot 3^{9}\cdot 13^{15}$
Root discriminant $31.72$
Ramified primes $2, 3, 13$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $C_2\times S_3^2$ (as 18T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![534, -2208, 1474, 1374, 1240, -1256, 1665, 1670, -1796, 1116, 361, -598, 400, -96, 3, 2, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 6*x^16 + 2*x^15 + 3*x^14 - 96*x^13 + 400*x^12 - 598*x^11 + 361*x^10 + 1116*x^9 - 1796*x^8 + 1670*x^7 + 1665*x^6 - 1256*x^5 + 1240*x^4 + 1374*x^3 + 1474*x^2 - 2208*x + 534)
 
gp: K = bnfinit(x^18 - 4*x^17 + 6*x^16 + 2*x^15 + 3*x^14 - 96*x^13 + 400*x^12 - 598*x^11 + 361*x^10 + 1116*x^9 - 1796*x^8 + 1670*x^7 + 1665*x^6 - 1256*x^5 + 1240*x^4 + 1374*x^3 + 1474*x^2 - 2208*x + 534, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 6 x^{16} + 2 x^{15} + 3 x^{14} - 96 x^{13} + 400 x^{12} - 598 x^{11} + 361 x^{10} + 1116 x^{9} - 1796 x^{8} + 1670 x^{7} + 1665 x^{6} - 1256 x^{5} + 1240 x^{4} + 1374 x^{3} + 1474 x^{2} - 2208 x + 534 \)

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:  \(-1056431860294718188453625856=-\,2^{20}\cdot 3^{9}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.72$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} + \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{12} + \frac{1}{12} a^{11} - \frac{1}{4} a^{10} - \frac{1}{12} a^{9} - \frac{1}{4} a^{8} - \frac{1}{12} a^{7} + \frac{1}{12} a^{6} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{72} a^{14} - \frac{1}{12} a^{12} + \frac{5}{36} a^{11} - \frac{2}{9} a^{10} - \frac{1}{18} a^{9} + \frac{13}{36} a^{8} + \frac{1}{72} a^{6} - \frac{1}{9} a^{5} + \frac{5}{36} a^{4} + \frac{1}{36} a^{3} + \frac{7}{36} a^{2} - \frac{5}{12}$, $\frac{1}{432} a^{15} - \frac{1}{432} a^{14} + \frac{1}{72} a^{13} - \frac{7}{216} a^{12} - \frac{43}{216} a^{11} - \frac{1}{72} a^{10} - \frac{5}{24} a^{9} + \frac{25}{108} a^{8} + \frac{169}{432} a^{7} + \frac{67}{144} a^{6} - \frac{1}{24} a^{5} - \frac{35}{108} a^{4} - \frac{1}{18} a^{3} + \frac{5}{216} a^{2} - \frac{5}{72} a + \frac{23}{72}$, $\frac{1}{42768} a^{16} - \frac{7}{14256} a^{15} - \frac{5}{5346} a^{14} - \frac{31}{21384} a^{13} + \frac{421}{21384} a^{12} - \frac{4513}{21384} a^{11} - \frac{521}{7128} a^{10} + \frac{1657}{10692} a^{9} + \frac{7577}{42768} a^{8} + \frac{20941}{42768} a^{7} + \frac{215}{1188} a^{6} - \frac{839}{10692} a^{5} + \frac{4327}{10692} a^{4} + \frac{7847}{21384} a^{3} + \frac{4247}{21384} a^{2} - \frac{607}{2376} a + \frac{1753}{3564}$, $\frac{1}{68202126800277175728} a^{17} - \frac{311681046419041}{34101063400138587864} a^{16} - \frac{4596065365807837}{4262632925017323483} a^{15} + \frac{105509509924535785}{22734042266759058576} a^{14} + \frac{121254013436223787}{3789007044459843096} a^{13} - \frac{451673642812962961}{11367021133379529288} a^{12} - \frac{5063900297670914125}{34101063400138587864} a^{11} + \frac{1844469689676854957}{8525265850034646966} a^{10} + \frac{4002300176932804025}{22734042266759058576} a^{9} + \frac{2651507300059539643}{5683510566689764644} a^{8} + \frac{1538053310928575437}{3100096672739871624} a^{7} + \frac{21942231271893045847}{68202126800277175728} a^{6} - \frac{12000530895898890791}{34101063400138587864} a^{5} - \frac{497786172279153893}{34101063400138587864} a^{4} - \frac{804306914134964131}{4262632925017323483} a^{3} + \frac{3126915711733022681}{34101063400138587864} a^{2} + \frac{904098984315889685}{2841755283344882322} a - \frac{2058962149723278697}{11367021133379529288}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( 4564456.230782871 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3^2$ (as 18T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 18 conjugacy class representatives for $C_2\times S_3^2$
Character table for $C_2\times S_3^2$

Intermediate fields

\(\Q(\sqrt{-39}) \), 3.1.2028.1, 3.1.676.1, 6.0.160398576.1, 6.0.160398576.2, 9.1.133451615232.2

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

Sibling fields

Degree 12 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
$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}$
13Data not computed