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Properties

Label 18.0.64777118927...9536.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 31^{9}\cdot 163^{14}$
Root discriminant $585.20$
Ramified primes $2, 31, 163$
Class number $2988090000$ (GRH)
Class group $[300, 9960300]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![105185281361, 3807287761, 25382044532, -1720750208, 3365343059, -102044671, 296925420, -7527429, 13778320, 1083121, 742416, 58411, 3114, -149, 1875, 6, -46, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 46*x^16 + 6*x^15 + 1875*x^14 - 149*x^13 + 3114*x^12 + 58411*x^11 + 742416*x^10 + 1083121*x^9 + 13778320*x^8 - 7527429*x^7 + 296925420*x^6 - 102044671*x^5 + 3365343059*x^4 - 1720750208*x^3 + 25382044532*x^2 + 3807287761*x + 105185281361)
 
gp: K = bnfinit(x^18 - x^17 - 46*x^16 + 6*x^15 + 1875*x^14 - 149*x^13 + 3114*x^12 + 58411*x^11 + 742416*x^10 + 1083121*x^9 + 13778320*x^8 - 7527429*x^7 + 296925420*x^6 - 102044671*x^5 + 3365343059*x^4 - 1720750208*x^3 + 25382044532*x^2 + 3807287761*x + 105185281361, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 46 x^{16} + 6 x^{15} + 1875 x^{14} - 149 x^{13} + 3114 x^{12} + 58411 x^{11} + 742416 x^{10} + 1083121 x^{9} + 13778320 x^{8} - 7527429 x^{7} + 296925420 x^{6} - 102044671 x^{5} + 3365343059 x^{4} - 1720750208 x^{3} + 25382044532 x^{2} + 3807287761 x + 105185281361 \)

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:  \(-64777118927098033917816196726443715275282899009536=-\,2^{18}\cdot 31^{9}\cdot 163^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $585.20$
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:  $2, 31, 163$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{74} a^{14} - \frac{7}{74} a^{13} - \frac{9}{74} a^{12} + \frac{33}{74} a^{11} - \frac{33}{74} a^{10} + \frac{3}{74} a^{9} + \frac{11}{74} a^{8} - \frac{21}{74} a^{7} - \frac{17}{74} a^{6} - \frac{29}{74} a^{5} - \frac{21}{74} a^{4} + \frac{7}{74} a^{3} + \frac{15}{74} a^{2} - \frac{21}{74} a + \frac{16}{37}$, $\frac{1}{1258} a^{15} + \frac{5}{1258} a^{14} - \frac{6}{37} a^{13} + \frac{295}{1258} a^{12} + \frac{237}{629} a^{11} - \frac{541}{1258} a^{10} - \frac{254}{629} a^{9} + \frac{5}{34} a^{8} - \frac{301}{629} a^{7} - \frac{455}{1258} a^{6} + \frac{167}{629} a^{5} - \frac{23}{1258} a^{4} - \frac{80}{629} a^{3} + \frac{381}{1258} a^{2} - \frac{331}{1258} a + \frac{155}{629}$, $\frac{1}{68929594} a^{16} - \frac{22395}{68929594} a^{15} + \frac{150941}{34464797} a^{14} + \frac{581216}{34464797} a^{13} + \frac{509907}{4054682} a^{12} + \frac{13248223}{34464797} a^{11} + \frac{33777065}{68929594} a^{10} - \frac{13476575}{34464797} a^{9} - \frac{1206815}{68929594} a^{8} - \frac{6217055}{34464797} a^{7} - \frac{28597161}{68929594} a^{6} - \frac{14581754}{34464797} a^{5} + \frac{30489525}{68929594} a^{4} - \frac{14957680}{34464797} a^{3} + \frac{1008234}{2027341} a^{2} + \frac{10456673}{68929594} a - \frac{17949375}{68929594}$, $\frac{1}{665938429392790246909978834315227422330961666029555776805586701147878} a^{17} + \frac{2221254627909909994598439445138687830703051580931955434681471}{332969214696395123454989417157613711165480833014777888402793350573939} a^{16} + \frac{37730710254691089879751098769511452941534771653731887506642791}{954066517754713820787935292715225533425446512936326327801700144911} a^{15} - \frac{403279187254324086619957222844184060383922241302042678303650512600}{332969214696395123454989417157613711165480833014777888402793350573939} a^{14} - \frac{96398290403365249354321798540779294502751479762857385041891112697815}{665938429392790246909978834315227422330961666029555776805586701147878} a^{13} + \frac{90841817511768893613018709577538949056397006494109768046777037783975}{665938429392790246909978834315227422330961666029555776805586701147878} a^{12} - \frac{301159118399760050845178932978884980642006125621135099808018749356641}{665938429392790246909978834315227422330961666029555776805586701147878} a^{11} - \frac{21460056685906035754322214898783684277304687888811198412270215570271}{665938429392790246909978834315227422330961666029555776805586701147878} a^{10} + \frac{233996267638308246658212119465450234451631269043973951767444178922427}{665938429392790246909978834315227422330961666029555776805586701147878} a^{9} + \frac{1893191251247550249776294471212204848650962713438007969649936792377}{4241646047087835967579483021116098231407399146685068642073800644254} a^{8} + \frac{9007191694617024374131897408514104455955522627728410278825051295221}{665938429392790246909978834315227422330961666029555776805586701147878} a^{7} + \frac{78106482322647988266379187923570771786714692727970540875683226434835}{665938429392790246909978834315227422330961666029555776805586701147878} a^{6} + \frac{40056665627796480098083364660102711907772773421265954966564758777671}{665938429392790246909978834315227422330961666029555776805586701147878} a^{5} + \frac{217387012781207271282960085400163380867421183765687269355140288448543}{665938429392790246909978834315227422330961666029555776805586701147878} a^{4} - \frac{159325805784037060252536004248149197228731629900013738770754014451405}{332969214696395123454989417157613711165480833014777888402793350573939} a^{3} - \frac{196677501659746031064234629636243481173965895766870804309325161914983}{665938429392790246909978834315227422330961666029555776805586701147878} a^{2} - \frac{193307324251394941410464447373682386184011277638781276081044657657199}{665938429392790246909978834315227422330961666029555776805586701147878} a + \frac{275002224875779894371002580492840358054821457203474984411799222117869}{665938429392790246909978834315227422330961666029555776805586701147878}$

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

Class group and class number

$C_{300}\times C_{9960300}$, which has order $2988090000$ (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:  \( 87986390.20265311 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-31}) \), 3.3.1304.1, 3.3.26569.1, 6.0.50657093056.2, 6.0.21029817271951.1, 9.9.1565248123502319104.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
$31$31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$163$163.3.2.1$x^{3} - 163$$3$$1$$2$$C_3$$[\ ]_{3}$
163.3.2.1$x^{3} - 163$$3$$1$$2$$C_3$$[\ ]_{3}$
163.6.5.5$x^{6} + 10432$$6$$1$$5$$C_6$$[\ ]_{6}$
163.6.5.5$x^{6} + 10432$$6$$1$$5$$C_6$$[\ ]_{6}$