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Properties

Label 18.12.9143056423...8672.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{12}\cdot 3^{24}\cdot 7^{13}\cdot 13^{8}$
Root discriminant $87.56$
Ramified primes $2, 3, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T585

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![448, 0, 27216, 0, -202104, 0, 17613, 0, 119094, 0, -55854, 0, 8372, 0, -309, 0, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 18*x^16 - 309*x^14 + 8372*x^12 - 55854*x^10 + 119094*x^8 + 17613*x^6 - 202104*x^4 + 27216*x^2 + 448)
 
gp: K = bnfinit(x^18 - 18*x^16 - 309*x^14 + 8372*x^12 - 55854*x^10 + 119094*x^8 + 17613*x^6 - 202104*x^4 + 27216*x^2 + 448, 1)
 

Normalized defining polynomial

\( x^{18} - 18 x^{16} - 309 x^{14} + 8372 x^{12} - 55854 x^{10} + 119094 x^{8} + 17613 x^{6} - 202104 x^{4} + 27216 x^{2} + 448 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-91430564233231293221640534671388672=-\,2^{12}\cdot 3^{24}\cdot 7^{13}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.56$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{14} - \frac{1}{4} a^{10} + \frac{1}{10} a^{8} - \frac{2}{5} a^{6} + \frac{2}{5} a^{4} + \frac{1}{4} a^{2} + \frac{2}{5}$, $\frac{1}{80} a^{15} + \frac{1}{8} a^{13} - \frac{1}{16} a^{11} - \frac{1}{10} a^{9} + \frac{1}{40} a^{7} - \frac{1}{2} a^{6} - \frac{1}{40} a^{5} - \frac{1}{2} a^{4} - \frac{7}{16} a^{3} + \frac{7}{20} a - \frac{1}{2}$, $\frac{1}{1515174663938706024800} a^{16} + \frac{1768336766781013239}{108226761709907573200} a^{14} - \frac{1}{4} a^{13} + \frac{6510153891466142887}{303034932787741204960} a^{12} - \frac{3427092378674055443}{94698416496169126550} a^{10} - \frac{1}{4} a^{9} - \frac{110681186334080336343}{757587331969353012400} a^{8} - \frac{52438809425696062141}{151517466393870602480} a^{6} - \frac{43619879599790393661}{216453523419815146400} a^{4} - \frac{1}{2} a^{3} - \frac{11868652521404125119}{54113380854953786600} a^{2} - \frac{1}{4} a + \frac{482398321809057932}{6764172606869223325}$, $\frac{1}{3030349327877412049600} a^{17} - \frac{937332275966676091}{216453523419815146400} a^{15} - \frac{1}{40} a^{14} + \frac{82268887088401444127}{606069865575482409920} a^{13} - \frac{1}{4} a^{12} + \frac{33640838733388341503}{757587331969353012400} a^{11} + \frac{1}{8} a^{10} + \frac{40836280059790266137}{1515174663938706024800} a^{9} + \frac{1}{5} a^{8} - \frac{12002936549077918453}{60606986557548240992} a^{7} - \frac{1}{20} a^{6} - \frac{141023965138707209541}{432907046839630292800} a^{5} + \frac{1}{20} a^{4} - \frac{2855594722234390109}{6764172606869223325} a^{3} - \frac{1}{8} a^{2} + \frac{5023300207739649859}{27056690427476893300} a + \frac{3}{10}$

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:  $14$
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:  \( 321270443912 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T585:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 13824
The 96 conjugacy class representatives for t18n585 are not computed
Character table for t18n585 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.1785733746591249.2

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 ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.4$x^{6} + x^{2} + 1$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
3Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.4.1$x^{6} + 39 x^{3} + 676$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$