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Properties

Label 18.0.34054756997...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 5^{15}\cdot 7^{12}$
Root discriminant $38.47$
Ramified primes $2, 3, 5, 7$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2\times C_3:S_3$ (as 18T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![216, -432, -144, 684, -240, -372, 634, -792, 982, -705, 206, 108, -179, 54, 50, -51, 22, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 22*x^16 - 51*x^15 + 50*x^14 + 54*x^13 - 179*x^12 + 108*x^11 + 206*x^10 - 705*x^9 + 982*x^8 - 792*x^7 + 634*x^6 - 372*x^5 - 240*x^4 + 684*x^3 - 144*x^2 - 432*x + 216)
 
gp: K = bnfinit(x^18 - 6*x^17 + 22*x^16 - 51*x^15 + 50*x^14 + 54*x^13 - 179*x^12 + 108*x^11 + 206*x^10 - 705*x^9 + 982*x^8 - 792*x^7 + 634*x^6 - 372*x^5 - 240*x^4 + 684*x^3 - 144*x^2 - 432*x + 216, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 22 x^{16} - 51 x^{15} + 50 x^{14} + 54 x^{13} - 179 x^{12} + 108 x^{11} + 206 x^{10} - 705 x^{9} + 982 x^{8} - 792 x^{7} + 634 x^{6} - 372 x^{5} - 240 x^{4} + 684 x^{3} - 144 x^{2} - 432 x + 216 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-34054756997160375000000000000=-\,2^{12}\cdot 3^{9}\cdot 5^{15}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.47$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{9} - \frac{1}{3} a^{8} - \frac{1}{2} a^{6} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{10} - \frac{1}{3} a^{9} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{11} - \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} + \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{24} a^{15} - \frac{1}{24} a^{12} - \frac{3}{8} a^{9} - \frac{5}{12} a^{8} + \frac{3}{8} a^{6} - \frac{1}{2} a^{5} - \frac{5}{12} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{5717016} a^{16} - \frac{5335}{317612} a^{15} + \frac{62615}{2858508} a^{14} - \frac{41639}{1905672} a^{13} + \frac{213475}{2858508} a^{12} - \frac{58597}{952836} a^{11} - \frac{812345}{5717016} a^{10} - \frac{27591}{79403} a^{9} + \frac{598249}{2858508} a^{8} + \frac{169385}{635224} a^{7} - \frac{156589}{2858508} a^{6} + \frac{69475}{158806} a^{5} - \frac{280537}{714627} a^{4} + \frac{7865}{238209} a^{3} - \frac{22166}{238209} a^{2} + \frac{72691}{158806} a + \frac{21088}{79403}$, $\frac{1}{456372236232} a^{17} - \frac{1723}{152124078744} a^{16} - \frac{226196614}{57046529529} a^{15} + \frac{3826942661}{152124078744} a^{14} - \frac{4510199779}{456372236232} a^{13} - \frac{981902473}{38031019686} a^{12} - \frac{34854436337}{456372236232} a^{11} + \frac{310391363}{13829461704} a^{10} + \frac{1656820411}{228186118116} a^{9} + \frac{6103097633}{13829461704} a^{8} + \frac{12513927055}{456372236232} a^{7} + \frac{31975499873}{76062039372} a^{6} + \frac{9927046429}{20744192556} a^{5} + \frac{2656129942}{6338503281} a^{4} - \frac{811175270}{2112834427} a^{3} - \frac{702232765}{6338503281} a^{2} + \frac{4134115379}{12677006562} a + \frac{38802591}{2112834427}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  $8$
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:  \( 15752633.920917612 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_3$ (as 18T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $C_2\times C_3:S_3$
Character table for $C_2\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.1.14700.1, 3.1.300.1, 3.1.3675.1, 3.1.588.1, 6.0.1350000.1, 6.0.3241350000.1, 6.0.202584375.1, 6.0.129654000.2, 9.1.9529569000000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: 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 R R R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.6.5.2$x^{6} + 10$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
7Data not computed