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Properties

Label 18.2.28865239569...1961.1
Degree $18$
Signature $[2, 8]$
Discriminant $7^{12}\cdot 53^{6}\cdot 97^{2}$
Root discriminant $22.85$
Ramified primes $7, 53, 97$
Class number $2$
Class group $[2]$
Galois group 18T463

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![512, 0, 256, -192, -128, -64, -48, 0, -8, -17, -4, 0, -6, -4, -4, -3, 2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 2*x^16 - 3*x^15 - 4*x^14 - 4*x^13 - 6*x^12 - 4*x^10 - 17*x^9 - 8*x^8 - 48*x^6 - 64*x^5 - 128*x^4 - 192*x^3 + 256*x^2 + 512)
 
gp: K = bnfinit(x^18 + 2*x^16 - 3*x^15 - 4*x^14 - 4*x^13 - 6*x^12 - 4*x^10 - 17*x^9 - 8*x^8 - 48*x^6 - 64*x^5 - 128*x^4 - 192*x^3 + 256*x^2 + 512, 1)
 

Normalized defining polynomial

\( x^{18} + 2 x^{16} - 3 x^{15} - 4 x^{14} - 4 x^{13} - 6 x^{12} - 4 x^{10} - 17 x^{9} - 8 x^{8} - 48 x^{6} - 64 x^{5} - 128 x^{4} - 192 x^{3} + 256 x^{2} + 512 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2886523956915912859821961=7^{12}\cdot 53^{6}\cdot 97^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.85$
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:  $7, 53, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{10} - \frac{3}{8} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{13} - \frac{1}{8} a^{11} - \frac{3}{16} a^{10} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{3}{8} a^{7} + \frac{1}{4} a^{5} - \frac{1}{16} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{14} - \frac{1}{16} a^{12} - \frac{3}{32} a^{11} + \frac{1}{8} a^{10} + \frac{1}{4} a^{9} + \frac{5}{16} a^{8} - \frac{3}{8} a^{6} + \frac{15}{32} a^{5} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{15} - \frac{1}{32} a^{13} - \frac{3}{64} a^{12} + \frac{1}{16} a^{11} + \frac{1}{8} a^{10} - \frac{11}{32} a^{9} - \frac{1}{2} a^{8} + \frac{5}{16} a^{7} - \frac{17}{64} a^{6} - \frac{3}{8} a^{5} + \frac{1}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{435584} a^{16} + \frac{537}{217792} a^{15} + \frac{1631}{217792} a^{14} + \frac{2953}{435584} a^{13} + \frac{10311}{217792} a^{12} + \frac{1839}{27224} a^{11} - \frac{3051}{217792} a^{10} - \frac{46883}{108896} a^{9} - \frac{29131}{108896} a^{8} - \frac{113033}{435584} a^{7} + \frac{4811}{217792} a^{6} + \frac{997}{108896} a^{5} - \frac{16607}{54448} a^{4} + \frac{2503}{27224} a^{3} + \frac{3121}{13612} a^{2} - \frac{1255}{6806} a + \frac{2}{3403}$, $\frac{1}{871168} a^{17} - \frac{1}{128} a^{15} + \frac{11461}{871168} a^{14} + \frac{1771}{217792} a^{13} - \frac{3179}{217792} a^{12} + \frac{54241}{435584} a^{11} - \frac{1839}{27224} a^{10} + \frac{67707}{217792} a^{9} - \frac{360001}{871168} a^{8} + \frac{12115}{108896} a^{7} + \frac{22303}{108896} a^{6} - \frac{10919}{108896} a^{5} + \frac{2301}{27224} a^{4} - \frac{26}{3403} a^{3} + \frac{447}{13612} a^{2} + \frac{143}{6806} a - \frac{1074}{3403}$

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

Class group and class number

$C_{2}$, which has order $2$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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:  \( 35079.4331442 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T463:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 48 conjugacy class representatives for t18n463
Character table for t18n463 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 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ 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}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
53.6.3.1$x^{6} - 106 x^{4} + 2809 x^{2} - 9528128$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$97$$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{97}$$x + 5$$1$$1$$0$Trivial$[\ ]$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
97.2.1.2$x^{2} + 485$$2$$1$$1$$C_2$$[\ ]_{2}$
97.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$