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Properties

Label 18.0.24254200053...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 5^{10}\cdot 23^{6}$
Root discriminant $17.52$
Ramified primes $2, 5, 23$
Class number $1$
Class group Trivial
Galois group $C_3:S_3:S_4$ (as 18T153)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 0, -75, 0, 175, 0, -130, 0, 150, 0, -25, 0, 47, 0, 9, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 4*x^16 + 9*x^14 + 47*x^12 - 25*x^10 + 150*x^8 - 130*x^6 + 175*x^4 - 75*x^2 + 25)
 
gp: K = bnfinit(x^18 + 4*x^16 + 9*x^14 + 47*x^12 - 25*x^10 + 150*x^8 - 130*x^6 + 175*x^4 - 75*x^2 + 25, 1)
 

Normalized defining polynomial

\( x^{18} + 4 x^{16} + 9 x^{14} + 47 x^{12} - 25 x^{10} + 150 x^{8} - 130 x^{6} + 175 x^{4} - 75 x^{2} + 25 \)

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:  \(-24254200053760000000000=-\,2^{24}\cdot 5^{10}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.52$
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:  $2, 5, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{40} a^{12} + \frac{1}{10} a^{10} + \frac{1}{10} a^{8} - \frac{1}{5} a^{6} + \frac{3}{8} a^{4} + \frac{1}{4} a^{2} - \frac{1}{8}$, $\frac{1}{40} a^{13} + \frac{1}{10} a^{11} + \frac{1}{10} a^{9} - \frac{1}{5} a^{7} + \frac{3}{8} a^{5} + \frac{1}{4} a^{3} - \frac{1}{8} a$, $\frac{1}{120} a^{14} - \frac{1}{120} a^{12} + \frac{1}{30} a^{10} - \frac{1}{15} a^{8} - \frac{5}{24} a^{6} + \frac{11}{24} a^{4} + \frac{5}{24} a^{2} - \frac{7}{24}$, $\frac{1}{240} a^{15} - \frac{1}{240} a^{13} - \frac{1}{80} a^{12} - \frac{7}{30} a^{11} - \frac{1}{20} a^{10} - \frac{1}{30} a^{9} - \frac{1}{20} a^{8} + \frac{7}{48} a^{7} - \frac{3}{20} a^{6} + \frac{23}{48} a^{5} + \frac{5}{16} a^{4} - \frac{19}{48} a^{3} - \frac{3}{8} a^{2} - \frac{7}{48} a + \frac{5}{16}$, $\frac{1}{163440} a^{16} + \frac{27}{18160} a^{14} - \frac{1}{80} a^{13} + \frac{49}{9080} a^{12} - \frac{1}{20} a^{11} + \frac{26}{10215} a^{10} - \frac{1}{20} a^{9} + \frac{8617}{54480} a^{8} - \frac{3}{20} a^{7} + \frac{2893}{54480} a^{6} + \frac{5}{16} a^{5} - \frac{6845}{32688} a^{4} - \frac{3}{8} a^{3} + \frac{4609}{32688} a^{2} + \frac{5}{16} a + \frac{5287}{16344}$, $\frac{1}{163440} a^{17} + \frac{27}{18160} a^{15} - \frac{1}{240} a^{14} + \frac{49}{9080} a^{13} - \frac{1}{120} a^{12} + \frac{26}{10215} a^{11} + \frac{11}{60} a^{10} + \frac{8617}{54480} a^{9} - \frac{1}{60} a^{8} + \frac{2893}{54480} a^{7} + \frac{49}{240} a^{6} - \frac{6845}{32688} a^{5} - \frac{1}{6} a^{4} + \frac{4609}{32688} a^{3} - \frac{23}{48} a^{2} + \frac{5287}{16344} a - \frac{1}{24}$

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:  $8$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 5357.32725649 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3:S_4$ (as 18T153):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 20 conjugacy class representatives for $C_3:S_3:S_4$
Character table for $C_3:S_3:S_4$

Intermediate fields

3.1.23.1, 6.0.846400.2, 9.1.486680000.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$2.6.6.4$x^{6} + x^{2} + 1$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.12.18.51$x^{12} + 10 x^{11} + 16 x^{10} + 16 x^{9} - 6 x^{8} + 16 x^{7} - 8 x^{6} - 8 x^{5} + 4 x^{4} - 8 x^{3} + 16 x + 8$$4$$3$$18$$A_4 \times C_2$$[2, 2, 2]^{3}$
$5$5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.12.10.4$x^{12} - 5 x^{6} + 50$$6$$2$$10$$C_3\times (C_3 : C_4)$$[\ ]_{6}^{6}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$