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Properties

Label 18.18.1156378877...0000.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 5^{15}\cdot 41^{13}$
Root discriminant $88.71$
Ramified primes $2, 5, 41$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $C_2\times C_3:S_3.C_2$ (as 18T27)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3120, -28960, -99600, -142160, -14440, 184160, 150504, -61340, -113412, -5760, 38470, 7640, -7075, -1705, 740, 155, -42, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 - 42*x^16 + 155*x^15 + 740*x^14 - 1705*x^13 - 7075*x^12 + 7640*x^11 + 38470*x^10 - 5760*x^9 - 113412*x^8 - 61340*x^7 + 150504*x^6 + 184160*x^5 - 14440*x^4 - 142160*x^3 - 99600*x^2 - 28960*x - 3120)
 
gp: K = bnfinit(x^18 - 5*x^17 - 42*x^16 + 155*x^15 + 740*x^14 - 1705*x^13 - 7075*x^12 + 7640*x^11 + 38470*x^10 - 5760*x^9 - 113412*x^8 - 61340*x^7 + 150504*x^6 + 184160*x^5 - 14440*x^4 - 142160*x^3 - 99600*x^2 - 28960*x - 3120, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} - 42 x^{16} + 155 x^{15} + 740 x^{14} - 1705 x^{13} - 7075 x^{12} + 7640 x^{11} + 38470 x^{10} - 5760 x^{9} - 113412 x^{8} - 61340 x^{7} + 150504 x^{6} + 184160 x^{5} - 14440 x^{4} - 142160 x^{3} - 99600 x^{2} - 28960 x - 3120 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(115637887789376703665125000000000000=2^{12}\cdot 5^{15}\cdot 41^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.71$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} + \frac{1}{8} a^{10} - \frac{3}{8} a^{8} - \frac{3}{8} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{14} + \frac{1}{24} a^{12} - \frac{1}{24} a^{11} + \frac{5}{24} a^{10} + \frac{1}{8} a^{9} + \frac{5}{12} a^{8} + \frac{5}{24} a^{7} + \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a$, $\frac{1}{48} a^{15} - \frac{1}{48} a^{14} - \frac{1}{24} a^{13} + \frac{1}{48} a^{12} - \frac{1}{8} a^{11} - \frac{5}{48} a^{10} - \frac{5}{48} a^{9} - \frac{1}{6} a^{8} - \frac{11}{24} a^{6} + \frac{1}{12} a^{4} - \frac{1}{4} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{336} a^{16} - \frac{1}{112} a^{14} - \frac{1}{48} a^{13} - \frac{5}{48} a^{12} - \frac{11}{336} a^{11} - \frac{1}{84} a^{10} + \frac{23}{336} a^{9} - \frac{25}{168} a^{8} - \frac{17}{42} a^{7} - \frac{5}{168} a^{6} + \frac{1}{21} a^{5} - \frac{19}{42} a^{4} - \frac{19}{84} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{14}$, $\frac{1}{51774550925664} a^{17} - \frac{6683798111}{51774550925664} a^{16} - \frac{87661122189}{8629091820944} a^{15} - \frac{279595436857}{51774550925664} a^{14} - \frac{6724752171}{616363701496} a^{13} + \frac{5022496085057}{51774550925664} a^{12} + \frac{2113198181287}{17258183641888} a^{11} + \frac{3149278658401}{12943637731416} a^{10} - \frac{998212426489}{8629091820944} a^{9} + \frac{424170457689}{8629091820944} a^{8} + \frac{510322268401}{3235909432854} a^{7} - \frac{64458912965}{154090925374} a^{6} - \frac{159612642985}{12943637731416} a^{5} - \frac{110295082159}{308181850748} a^{4} - \frac{268733002789}{2157272955236} a^{3} - \frac{137228295329}{308181850748} a^{2} - \frac{1205070702187}{3235909432854} a + \frac{397646816229}{1078636477618}$

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:  $17$
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:  \( 799852501138 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_3.C_2$ (as 18T27):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 12 conjugacy class representatives for $C_2\times C_3:S_3.C_2$
Character table for $C_2\times C_3:S_3.C_2$

Intermediate fields

\(\Q(\sqrt{205}) \), 9.9.4750104241000000.1

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

Sibling fields

Degree 12 siblings: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$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}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$5$5.6.5.1$x^{6} - 5$$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$$[\ ]_{6}^{2}$
$41$41.2.1.1$x^{2} - 41$$2$$1$$1$$C_2$$[\ ]_{2}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.1$x^{4} - 41$$4$$1$$3$$C_4$$[\ ]_{4}$