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Properties

Label 18.6.31773322854...0000.3
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 3^{26}\cdot 5^{15}$
Root discriminant $29.67$
Ramified primes $2, 3, 5$
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![9, 81, 0, -477, 585, -459, 459, 270, -333, 285, -180, 0, -80, 45, 0, -12, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 - 12*x^15 + 45*x^13 - 80*x^12 - 180*x^10 + 285*x^9 - 333*x^8 + 270*x^7 + 459*x^6 - 459*x^5 + 585*x^4 - 477*x^3 + 81*x + 9)
 
gp: K = bnfinit(x^18 - 3*x^16 - 12*x^15 + 45*x^13 - 80*x^12 - 180*x^10 + 285*x^9 - 333*x^8 + 270*x^7 + 459*x^6 - 459*x^5 + 585*x^4 - 477*x^3 + 81*x + 9, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{16} - 12 x^{15} + 45 x^{13} - 80 x^{12} - 180 x^{10} + 285 x^{9} - 333 x^{8} + 270 x^{7} + 459 x^{6} - 459 x^{5} + 585 x^{4} - 477 x^{3} + 81 x + 9 \)

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:  \(317733228541125000000000000=2^{12}\cdot 3^{26}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.67$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{15} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{7} + \frac{1}{15} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{15} a^{13} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} - \frac{2}{5} a^{8} - \frac{1}{3} a^{7} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{3} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{15} a^{14} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} - \frac{1}{3} a^{8} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5}$, $\frac{1}{15} a^{15} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{3} a^{9} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{15} a^{16} + \frac{1}{5} a^{11} - \frac{2}{15} a^{10} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{2449267516897104545025} a^{17} - \frac{28307344303267418701}{2449267516897104545025} a^{16} + \frac{9178342590839484136}{816422505632368181675} a^{15} - \frac{3883951771613762209}{489853503379420909005} a^{14} - \frac{16262676235246484939}{489853503379420909005} a^{13} - \frac{3131988162842459813}{489853503379420909005} a^{12} - \frac{44304782905700493478}{489853503379420909005} a^{11} - \frac{20056467921410873798}{489853503379420909005} a^{10} - \frac{64370584055936124799}{163284501126473636335} a^{9} - \frac{190238352711575342308}{489853503379420909005} a^{8} - \frac{846175788425287774813}{2449267516897104545025} a^{7} - \frac{718617791264355915967}{2449267516897104545025} a^{6} + \frac{38518226313515821967}{816422505632368181675} a^{5} - \frac{70192074706771609706}{163284501126473636335} a^{4} + \frac{53253876329766033377}{163284501126473636335} a^{3} + \frac{30678235491132026826}{816422505632368181675} a^{2} + \frac{179751379919086294874}{816422505632368181675} a - \frac{160468445680785761267}{816422505632368181675}$

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:  \( 4100877.0443168166 \) (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, 9.3.1594323000000.3

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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$3.6.8.6$x^{6} + 18 x^{2} + 36$$3$$2$$8$$C_6$$[2]^{2}$
3.12.18.76$x^{12} + 45 x^{11} - 33 x^{10} + 6 x^{9} - 72 x^{8} + 99 x^{7} - 39 x^{6} + 27 x^{5} + 9 x^{4} - 63 x^{3} - 81 x^{2} - 54 x - 72$$6$$2$$18$$C_6\times S_3$$[3/2, 2]_{2}^{2}$
$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}$