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Properties

Label 18.6.95704599043...5056.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 29\cdot 41^{8}$
Root discriminant $52.62$
Ramified primes $2, 3, 7, 29, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T657

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-533, 3813, -6478, 7078, 5393, -13591, 4766, -3235, -100, 1140, -1214, 1300, -250, 133, 97, -27, 22, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 22*x^16 - 27*x^15 + 97*x^14 + 133*x^13 - 250*x^12 + 1300*x^11 - 1214*x^10 + 1140*x^9 - 100*x^8 - 3235*x^7 + 4766*x^6 - 13591*x^5 + 5393*x^4 + 7078*x^3 - 6478*x^2 + 3813*x - 533)
 
gp: K = bnfinit(x^18 - 3*x^17 + 22*x^16 - 27*x^15 + 97*x^14 + 133*x^13 - 250*x^12 + 1300*x^11 - 1214*x^10 + 1140*x^9 - 100*x^8 - 3235*x^7 + 4766*x^6 - 13591*x^5 + 5393*x^4 + 7078*x^3 - 6478*x^2 + 3813*x - 533, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 22 x^{16} - 27 x^{15} + 97 x^{14} + 133 x^{13} - 250 x^{12} + 1300 x^{11} - 1214 x^{10} + 1140 x^{9} - 100 x^{8} - 3235 x^{7} + 4766 x^{6} - 13591 x^{5} + 5393 x^{4} + 7078 x^{3} - 6478 x^{2} + 3813 x - 533 \)

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:  \(9570459904352696272673858605056=2^{12}\cdot 3^{6}\cdot 7^{12}\cdot 29\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.62$
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:  $2, 3, 7, 29, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{14} a^{15} + \frac{1}{14} a^{14} + \frac{1}{7} a^{13} + \frac{5}{14} a^{12} - \frac{1}{7} a^{11} - \frac{2}{7} a^{10} - \frac{3}{7} a^{9} - \frac{5}{14} a^{8} + \frac{3}{7} a^{7} + \frac{1}{7} a^{6} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3} - \frac{3}{14} a^{2} + \frac{1}{14} a + \frac{3}{7}$, $\frac{1}{28} a^{16} - \frac{1}{28} a^{15} - \frac{1}{4} a^{14} - \frac{3}{14} a^{13} + \frac{1}{14} a^{12} + \frac{1}{4} a^{11} + \frac{1}{14} a^{10} + \frac{1}{4} a^{9} + \frac{1}{14} a^{8} + \frac{11}{28} a^{7} - \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{3}{14} a^{4} + \frac{9}{28} a^{3} - \frac{1}{4} a^{2} - \frac{3}{28} a + \frac{9}{28}$, $\frac{1}{666069181566777342128998161170708} a^{17} - \frac{1141952941686480429191353856310}{166517295391694335532249540292677} a^{16} + \frac{4739745792709313890589245340027}{333034590783388671064499080585354} a^{15} + \frac{133953563371462206137147630865469}{666069181566777342128998161170708} a^{14} - \frac{480655410484882193494573134591}{166517295391694335532249540292677} a^{13} - \frac{195784124451392749698714983833541}{666069181566777342128998161170708} a^{12} - \frac{124127972279514814766700166963391}{666069181566777342128998161170708} a^{11} - \frac{9309730293453511366809887334301}{95152740223825334589856880167244} a^{10} - \frac{2616992337539186704303259469729}{95152740223825334589856880167244} a^{9} - \frac{316280087242149047200123444763393}{666069181566777342128998161170708} a^{8} + \frac{261259222376437089332581737046163}{666069181566777342128998161170708} a^{7} - \frac{25343460228537494938017202673722}{166517295391694335532249540292677} a^{6} + \frac{112743354900466214743932531006681}{333034590783388671064499080585354} a^{5} - \frac{74993977456784646876953471003369}{666069181566777342128998161170708} a^{4} - \frac{153799084437554623429646544126249}{333034590783388671064499080585354} a^{3} - \frac{66651021183036892294227043504850}{166517295391694335532249540292677} a^{2} + \frac{22080752909312746596101570311374}{166517295391694335532249540292677} a - \frac{160590299376872235145954082427995}{666069181566777342128998161170708}$

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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 589606800.364 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T657:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27648
The 96 conjugacy class representatives for t18n657 are not computed
Character table for t18n657 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.574470067776192.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
41Data not computed