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Properties

Label 18.6.59273599077...8209.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 53^{6}\cdot 139^{2}$
Root discriminant $23.78$
Ramified primes $7, 53, 139$
Class number $1$
Class group Trivial
Galois group 18T269

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![127, -208, 225, -386, 183, 494, -213, 689, -356, 101, -278, 22, -34, 29, 32, 3, -8, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 8*x^16 + 3*x^15 + 32*x^14 + 29*x^13 - 34*x^12 + 22*x^11 - 278*x^10 + 101*x^9 - 356*x^8 + 689*x^7 - 213*x^6 + 494*x^5 + 183*x^4 - 386*x^3 + 225*x^2 - 208*x + 127)
 
gp: K = bnfinit(x^18 - 2*x^17 - 8*x^16 + 3*x^15 + 32*x^14 + 29*x^13 - 34*x^12 + 22*x^11 - 278*x^10 + 101*x^9 - 356*x^8 + 689*x^7 - 213*x^6 + 494*x^5 + 183*x^4 - 386*x^3 + 225*x^2 - 208*x + 127, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 8 x^{16} + 3 x^{15} + 32 x^{14} + 29 x^{13} - 34 x^{12} + 22 x^{11} - 278 x^{10} + 101 x^{9} - 356 x^{8} + 689 x^{7} - 213 x^{6} + 494 x^{5} + 183 x^{4} - 386 x^{3} + 225 x^{2} - 208 x + 127 \)

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:  \(5927359907702450033438209=7^{12}\cdot 53^{6}\cdot 139^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.78$
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:  $7, 53, 139$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} + \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{17427365338449159844109188} a^{17} - \frac{417636621794278805235669}{17427365338449159844109188} a^{16} + \frac{71151822382801750468467}{4356841334612289961027297} a^{15} - \frac{368207139714779147593064}{4356841334612289961027297} a^{14} - \frac{284062302263655057906355}{8713682669224579922054594} a^{13} + \frac{6354826322107558640770255}{17427365338449159844109188} a^{12} + \frac{131662277287051696524291}{17427365338449159844109188} a^{11} + \frac{1495460837555887084578361}{4356841334612289961027297} a^{10} - \frac{4516784386082890869172445}{17427365338449159844109188} a^{9} - \frac{4027404769465893559954209}{8713682669224579922054594} a^{8} - \frac{8489590007025484916592959}{17427365338449159844109188} a^{7} - \frac{2993489419226526193984253}{8713682669224579922054594} a^{6} - \frac{3562734267324797051187189}{8713682669224579922054594} a^{5} - \frac{1660018454216606808569989}{4356841334612289961027297} a^{4} + \frac{3467300052593072320495065}{17427365338449159844109188} a^{3} + \frac{1428082656523879747888579}{17427365338449159844109188} a^{2} - \frac{606083544398876491296431}{17427365338449159844109188} a + \frac{3982032671209502230085793}{8713682669224579922054594}$

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:  \( 181084.693245 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T269:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1152
The 24 conjugacy class representatives for t18n269
Character table for t18n269 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.2597.1, 9.9.17515230173.1

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 sibling: 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$53$53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.12.6.1$x^{12} + 2382032 x^{6} - 418195493 x^{2} + 1418519112256$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$139$$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{139}$$x + 4$$1$$1$$0$Trivial$[\ ]$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
139.4.2.1$x^{4} + 417 x^{2} + 77284$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
139.4.0.1$x^{4} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$