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Properties

Label 18.0.21423624263...6112.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{26}\cdot 3^{9}\cdot 7^{6}\cdot 13^{10}$
Root discriminant $37.49$
Ramified primes $2, 3, 7, 13$
Class number $3$ (GRH)
Class group $[3]$ (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![34300, -143080, 217588, -32494, -276628, 253950, 51751, -184794, 93111, -12520, 650, -2154, 677, 24, -72, 36, 3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 3*x^16 + 36*x^15 - 72*x^14 + 24*x^13 + 677*x^12 - 2154*x^11 + 650*x^10 - 12520*x^9 + 93111*x^8 - 184794*x^7 + 51751*x^6 + 253950*x^5 - 276628*x^4 - 32494*x^3 + 217588*x^2 - 143080*x + 34300)
 
gp: K = bnfinit(x^18 - 6*x^17 + 3*x^16 + 36*x^15 - 72*x^14 + 24*x^13 + 677*x^12 - 2154*x^11 + 650*x^10 - 12520*x^9 + 93111*x^8 - 184794*x^7 + 51751*x^6 + 253950*x^5 - 276628*x^4 - 32494*x^3 + 217588*x^2 - 143080*x + 34300, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 3 x^{16} + 36 x^{15} - 72 x^{14} + 24 x^{13} + 677 x^{12} - 2154 x^{11} + 650 x^{10} - 12520 x^{9} + 93111 x^{8} - 184794 x^{7} + 51751 x^{6} + 253950 x^{5} - 276628 x^{4} - 32494 x^{3} + 217588 x^{2} - 143080 x + 34300 \)

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:  \(-21423624263414745793258586112=-\,2^{26}\cdot 3^{9}\cdot 7^{6}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.49$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{35} a^{13} - \frac{3}{35} a^{12} - \frac{9}{35} a^{11} - \frac{9}{35} a^{10} + \frac{16}{35} a^{9} + \frac{6}{35} a^{8} - \frac{13}{35} a^{7} - \frac{11}{35} a^{6} + \frac{11}{35} a^{5} + \frac{2}{5} a^{4} + \frac{16}{35} a^{3} - \frac{12}{35} a^{2} - \frac{2}{5} a$, $\frac{1}{35} a^{14} + \frac{17}{35} a^{12} - \frac{1}{35} a^{11} - \frac{11}{35} a^{10} - \frac{16}{35} a^{9} + \frac{1}{7} a^{8} - \frac{3}{7} a^{7} + \frac{13}{35} a^{6} + \frac{12}{35} a^{5} - \frac{12}{35} a^{4} + \frac{1}{35} a^{3} - \frac{3}{7} a^{2} - \frac{1}{5} a$, $\frac{1}{70} a^{15} - \frac{1}{70} a^{13} + \frac{9}{35} a^{12} - \frac{12}{35} a^{11} + \frac{3}{35} a^{10} - \frac{3}{70} a^{9} - \frac{9}{35} a^{8} + \frac{1}{35} a^{7} - \frac{1}{2} a^{5} - \frac{3}{35} a^{4} - \frac{23}{70} a^{3} + \frac{17}{35} a^{2} - \frac{2}{5} a$, $\frac{1}{490} a^{16} + \frac{1}{245} a^{15} + \frac{1}{98} a^{14} + \frac{3}{245} a^{13} + \frac{37}{245} a^{12} - \frac{1}{5} a^{11} + \frac{33}{490} a^{10} + \frac{108}{245} a^{8} - \frac{48}{245} a^{7} + \frac{153}{490} a^{6} - \frac{92}{245} a^{5} - \frac{107}{490} a^{4} - \frac{48}{245} a^{3} + \frac{1}{7} a^{2} - \frac{2}{35} a$, $\frac{1}{5195002178222079259474666403275703890} a^{17} - \frac{5063218262051060324420162273344617}{5195002178222079259474666403275703890} a^{16} - \frac{36277168709427471086558082468534857}{5195002178222079259474666403275703890} a^{15} - \frac{14632926964648806584761965627471127}{1039000435644415851894933280655140778} a^{14} - \frac{37028302206879856247289204045746626}{2597501089111039629737333201637851945} a^{13} - \frac{82608511783212910082408470754464621}{519500217822207925947466640327570389} a^{12} - \frac{149985626733730225378906547253601099}{5195002178222079259474666403275703890} a^{11} + \frac{1336649012039549332088952677150591539}{5195002178222079259474666403275703890} a^{10} - \frac{620432437915381203280747180177138126}{2597501089111039629737333201637851945} a^{9} + \frac{166055216415163333735879843880959334}{371071584158719947105333314519693135} a^{8} + \frac{979620949992726052925454198589391407}{5195002178222079259474666403275703890} a^{7} + \frac{31547839989676829319660573972360879}{98018909023058099235371064212749130} a^{6} - \frac{990810333196568501923462783943265599}{5195002178222079259474666403275703890} a^{5} + \frac{303310385312445379283839672453604929}{5195002178222079259474666403275703890} a^{4} + \frac{810249214422393117054110141840300932}{2597501089111039629737333201637851945} a^{3} + \frac{139479099272614053690147510129349857}{371071584158719947105333314519693135} a^{2} - \frac{28194974368694010647534016868571331}{74214316831743989421066662903938627} a + \frac{1877945757631697385188200415697060}{10602045261677712774438094700562661}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -\frac{496420080872530195347761}{134848137328992242090019970} a^{17} + \frac{1157413271139649587517433}{67424068664496121045009985} a^{16} + \frac{818540017525873407977846}{67424068664496121045009985} a^{15} - \frac{7898152328111331121651218}{67424068664496121045009985} a^{14} + \frac{14401135879399958821435597}{134848137328992242090019970} a^{13} + \frac{4055843878580149773105588}{67424068664496121045009985} a^{12} - \frac{325369432107158936268641837}{134848137328992242090019970} a^{11} + \frac{316808509812809483286727682}{67424068664496121045009985} a^{10} + \frac{108907746652542353083014165}{26969627465798448418003994} a^{9} + \frac{692391939408725994949483277}{13484813732899224209001997} a^{8} - \frac{37019154862225585029674522001}{134848137328992242090019970} a^{7} + \frac{393863627550981434291005564}{1272152238952757000849245} a^{6} + \frac{15976867477599680789995031567}{67424068664496121045009985} a^{5} - \frac{8410412741412790607063356313}{13484813732899224209001997} a^{4} + \frac{22023715586511625996090874653}{134848137328992242090019970} a^{3} + \frac{24135168416560644590280382662}{67424068664496121045009985} a^{2} - \frac{21236073514182469164529569514}{67424068664496121045009985} a + \frac{1216715851185466286411991180}{13484813732899224209001997} \) (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:  \( 21541865.688527487 \) (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

\(\Q(\sqrt{-3}) \), 3.1.728.1, 6.0.73008.1, 6.0.14309568.1

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.12.22.27$x^{12} + 2 x^{6} + 4$$6$$2$$22$$C_6\times S_3$$[3]_{3}^{6}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.3$x^{3} - 52$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.5.1$x^{6} - 52$$6$$1$$5$$C_6$$[\ ]_{6}$