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Properties

Label 18.18.3848305294...0069.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{3}\cdot 7^{3}\cdot 229^{6}\cdot 257^{6}$
Root discriminant $64.61$
Ramified primes $3, 7, 229, 257$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times S_3\times S_4$ (as 18T111)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, 336, 1538, -12773, 14662, 25842, -46841, -13506, 48483, -4234, -22289, 5850, 4683, -1830, -366, 222, -3, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 - 3*x^16 + 222*x^15 - 366*x^14 - 1830*x^13 + 4683*x^12 + 5850*x^11 - 22289*x^10 - 4234*x^9 + 48483*x^8 - 13506*x^7 - 46841*x^6 + 25842*x^5 + 14662*x^4 - 12773*x^3 + 1538*x^2 + 336*x - 7)
 
gp: K = bnfinit(x^18 - 9*x^17 - 3*x^16 + 222*x^15 - 366*x^14 - 1830*x^13 + 4683*x^12 + 5850*x^11 - 22289*x^10 - 4234*x^9 + 48483*x^8 - 13506*x^7 - 46841*x^6 + 25842*x^5 + 14662*x^4 - 12773*x^3 + 1538*x^2 + 336*x - 7, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} - 3 x^{16} + 222 x^{15} - 366 x^{14} - 1830 x^{13} + 4683 x^{12} + 5850 x^{11} - 22289 x^{10} - 4234 x^{9} + 48483 x^{8} - 13506 x^{7} - 46841 x^{6} + 25842 x^{5} + 14662 x^{4} - 12773 x^{3} + 1538 x^{2} + 336 x - 7 \)

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:  \(384830529416792619956400279450069=3^{3}\cdot 7^{3}\cdot 229^{6}\cdot 257^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.61$
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:  $3, 7, 229, 257$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{35647757374} a^{16} - \frac{932656924}{17823878687} a^{15} - \frac{4319135485}{35647757374} a^{14} - \frac{180500057}{35647757374} a^{13} + \frac{1471791401}{17823878687} a^{12} - \frac{3549943045}{35647757374} a^{11} + \frac{1135091815}{35647757374} a^{10} + \frac{1355486115}{17823878687} a^{9} - \frac{2154151775}{17823878687} a^{8} + \frac{6720836477}{35647757374} a^{7} + \frac{11373393801}{35647757374} a^{6} + \frac{4089702235}{17823878687} a^{5} - \frac{6719688379}{35647757374} a^{4} + \frac{3844580573}{35647757374} a^{3} + \frac{7238119283}{35647757374} a^{2} + \frac{7122860}{109348949} a - \frac{8629261429}{17823878687}$, $\frac{1}{268391965268846} a^{17} + \frac{1349}{268391965268846} a^{16} - \frac{49355255497751}{268391965268846} a^{15} + \frac{45749810620429}{268391965268846} a^{14} + \frac{31863775057599}{134195982634423} a^{13} + \frac{1534110095109}{134195982634423} a^{12} + \frac{78201323108939}{268391965268846} a^{11} + \frac{62291486660003}{134195982634423} a^{10} + \frac{34428595103249}{268391965268846} a^{9} + \frac{7301449815976}{134195982634423} a^{8} - \frac{127611868324341}{268391965268846} a^{7} + \frac{14630859899207}{268391965268846} a^{6} + \frac{19446961854001}{268391965268846} a^{5} - \frac{2915159192986}{134195982634423} a^{4} - \frac{40259112723328}{134195982634423} a^{3} - \frac{127190626053955}{268391965268846} a^{2} - \frac{41838259798593}{134195982634423} a - \frac{116071696156203}{268391965268846}$

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:  $17$
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:  \( 35328403808.9 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3\times S_4$ (as 18T111):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 288
The 30 conjugacy class representatives for $C_2\times S_3\times S_4$
Character table for $C_2\times S_3\times S_4$ is not computed

Intermediate fields

3.3.229.1, 3.3.257.1, 6.6.1387029.1, 9.9.203847700616477.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
229Data not computed
257Data not computed