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Properties

Label 18.6.19533453031...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 3^{6}\cdot 5^{15}\cdot 11^{8}$
Root discriminant $25.41$
Ramified primes $2, 3, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_3^2:S_3$ (as 18T52)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-449, -584, -504, 3221, -6140, 7046, -3924, 136, 1926, -1955, 836, -24, -204, 156, -50, 1, 6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 6*x^16 + x^15 - 50*x^14 + 156*x^13 - 204*x^12 - 24*x^11 + 836*x^10 - 1955*x^9 + 1926*x^8 + 136*x^7 - 3924*x^6 + 7046*x^5 - 6140*x^4 + 3221*x^3 - 504*x^2 - 584*x - 449)
 
gp: K = bnfinit(x^18 - 4*x^17 + 6*x^16 + x^15 - 50*x^14 + 156*x^13 - 204*x^12 - 24*x^11 + 836*x^10 - 1955*x^9 + 1926*x^8 + 136*x^7 - 3924*x^6 + 7046*x^5 - 6140*x^4 + 3221*x^3 - 504*x^2 - 584*x - 449, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 6 x^{16} + x^{15} - 50 x^{14} + 156 x^{13} - 204 x^{12} - 24 x^{11} + 836 x^{10} - 1955 x^{9} + 1926 x^{8} + 136 x^{7} - 3924 x^{6} + 7046 x^{5} - 6140 x^{4} + 3221 x^{3} - 504 x^{2} - 584 x - 449 \)

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:  \(19533453031125000000000000=2^{12}\cdot 3^{6}\cdot 5^{15}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.41$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{35354904364864187986502381} a^{17} - \frac{647721656940533888441662}{2079700256756716940382493} a^{16} + \frac{8083254623442523644946260}{35354904364864187986502381} a^{15} + \frac{8321095567540186387578530}{35354904364864187986502381} a^{14} - \frac{11000089618310160812182463}{35354904364864187986502381} a^{13} - \frac{15962604209653879582902148}{35354904364864187986502381} a^{12} + \frac{12948051248937882331741045}{35354904364864187986502381} a^{11} + \frac{8001372560603354534475144}{35354904364864187986502381} a^{10} + \frac{14523513783148078953347953}{35354904364864187986502381} a^{9} - \frac{10013960397088404097783337}{35354904364864187986502381} a^{8} - \frac{13422462198241626645277004}{35354904364864187986502381} a^{7} - \frac{11719885561637053842258216}{35354904364864187986502381} a^{6} - \frac{16693826309602682843871816}{35354904364864187986502381} a^{5} - \frac{13977227476152694261778828}{35354904364864187986502381} a^{4} + \frac{10558332034174363239475587}{35354904364864187986502381} a^{3} + \frac{536812926702503065966287}{1219134633271178896086289} a^{2} - \frac{2267239652653409493045319}{35354904364864187986502381} a - \frac{9068425818610440387583444}{35354904364864187986502381}$

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:  \( 834349.6588433163 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2:S_3$ (as 18T52):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times C_3^2:S_3$
Character table for $C_2\times C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.1.300.1, 6.2.450000.1, 9.3.395307000000.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
2Data not computed
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
$11$11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$