↑

Properties

Label 18.6.64130955524...4848.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{9}\cdot 3^{6}\cdot 107^{8}$
Root discriminant $16.27$
Ramified primes $2, 3, 107$
Class number $1$
Class group Trivial
Galois group $C_2\times S_3\times S_4$ (as 18T111)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 6, 21, -62, 72, -82, 27, 52, -36, 39, -20, -28, 4, 10, 7, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 5*x^16 + 7*x^15 + 10*x^14 + 4*x^13 - 28*x^12 - 20*x^11 + 39*x^10 - 36*x^9 + 52*x^8 + 27*x^7 - 82*x^6 + 72*x^5 - 62*x^4 + 21*x^3 + 6*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^18 - 2*x^17 - 5*x^16 + 7*x^15 + 10*x^14 + 4*x^13 - 28*x^12 - 20*x^11 + 39*x^10 - 36*x^9 + 52*x^8 + 27*x^7 - 82*x^6 + 72*x^5 - 62*x^4 + 21*x^3 + 6*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 5 x^{16} + 7 x^{15} + 10 x^{14} + 4 x^{13} - 28 x^{12} - 20 x^{11} + 39 x^{10} - 36 x^{9} + 52 x^{8} + 27 x^{7} - 82 x^{6} + 72 x^{5} - 62 x^{4} + 21 x^{3} + 6 x^{2} - 4 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6413095552499045134848=2^{9}\cdot 3^{6}\cdot 107^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.27$
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:  $2, 3, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{129365753063} a^{17} - \frac{13313985168}{129365753063} a^{16} + \frac{10789454044}{129365753063} a^{15} - \frac{62104875593}{129365753063} a^{14} - \frac{19535993847}{129365753063} a^{13} + \frac{12115368957}{129365753063} a^{12} + \frac{49354498788}{129365753063} a^{11} - \frac{40461227135}{129365753063} a^{10} + \frac{29539649955}{129365753063} a^{9} + \frac{8058662840}{129365753063} a^{8} + \frac{55551837120}{129365753063} a^{7} + \frac{39623702169}{129365753063} a^{6} + \frac{16811023918}{129365753063} a^{5} + \frac{52861711067}{129365753063} a^{4} + \frac{21524879132}{129365753063} a^{3} - \frac{3981651117}{129365753063} a^{2} - \frac{48068758510}{129365753063} a - \frac{51856928273}{129365753063}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 7673.59396552 \)
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

3.3.321.1, 3.1.107.1, 6.2.91592.1, 9.3.3539149227.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.12.0.1$x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$107$107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$