↑

Properties

Label 18.18.8579614079...9984.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 3^{25}\cdot 47^{11}$
Root discriminant $76.77$
Ramified primes $2, 3, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T217

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6561, -78732, 223074, 23814, -574938, 148230, 603459, -153702, -332226, 57000, 100266, -8910, -16313, 594, 1404, -14, -60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 60*x^16 - 14*x^15 + 1404*x^14 + 594*x^13 - 16313*x^12 - 8910*x^11 + 100266*x^10 + 57000*x^9 - 332226*x^8 - 153702*x^7 + 603459*x^6 + 148230*x^5 - 574938*x^4 + 23814*x^3 + 223074*x^2 - 78732*x + 6561)
 
gp: K = bnfinit(x^18 - 60*x^16 - 14*x^15 + 1404*x^14 + 594*x^13 - 16313*x^12 - 8910*x^11 + 100266*x^10 + 57000*x^9 - 332226*x^8 - 153702*x^7 + 603459*x^6 + 148230*x^5 - 574938*x^4 + 23814*x^3 + 223074*x^2 - 78732*x + 6561, 1)
 

Normalized defining polynomial

\( x^{18} - 60 x^{16} - 14 x^{15} + 1404 x^{14} + 594 x^{13} - 16313 x^{12} - 8910 x^{11} + 100266 x^{10} + 57000 x^{9} - 332226 x^{8} - 153702 x^{7} + 603459 x^{6} + 148230 x^{5} - 574938 x^{4} + 23814 x^{3} + 223074 x^{2} - 78732 x + 6561 \)

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:  \(8579614079673266727444373410729984=2^{12}\cdot 3^{25}\cdot 47^{11}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.77$
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:  $2, 3, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6} a^{9} + \frac{1}{6} a^{6} + \frac{1}{6} a^{3} - \frac{1}{2}$, $\frac{1}{18} a^{10} + \frac{7}{18} a^{7} + \frac{1}{3} a^{5} + \frac{7}{18} a^{4} - \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{18} a^{11} + \frac{7}{18} a^{8} + \frac{1}{3} a^{6} + \frac{7}{18} a^{5} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{18} a^{12} + \frac{1}{18} a^{9} + \frac{1}{3} a^{7} + \frac{1}{18} a^{6} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{54} a^{13} + \frac{1}{54} a^{10} - \frac{2}{9} a^{8} + \frac{19}{54} a^{7} - \frac{1}{3} a^{6} + \frac{2}{9} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{324} a^{14} - \frac{1}{108} a^{13} + \frac{1}{108} a^{12} + \frac{1}{81} a^{11} + \frac{1}{54} a^{10} - \frac{1}{18} a^{9} + \frac{65}{162} a^{8} - \frac{4}{27} a^{7} + \frac{2}{27} a^{6} - \frac{7}{54} a^{5} - \frac{4}{9} a^{4} + \frac{4}{9} a^{3} - \frac{5}{36} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{10692} a^{15} - \frac{1}{891} a^{14} - \frac{5}{891} a^{13} + \frac{247}{10692} a^{12} + \frac{5}{891} a^{11} - \frac{1}{99} a^{10} - \frac{367}{5346} a^{9} - \frac{94}{891} a^{8} + \frac{8}{891} a^{7} + \frac{46}{891} a^{6} - \frac{20}{99} a^{5} + \frac{109}{297} a^{4} + \frac{61}{1188} a^{3} - \frac{28}{99} a^{2} + \frac{14}{33} a + \frac{3}{44}$, $\frac{1}{32076} a^{16} - \frac{1}{5346} a^{14} + \frac{11}{2916} a^{13} - \frac{5}{297} a^{12} + \frac{5}{198} a^{11} + \frac{173}{16038} a^{10} + \frac{1}{198} a^{9} + \frac{763}{5346} a^{8} - \frac{848}{2673} a^{7} - \frac{709}{1782} a^{6} - \frac{529}{1782} a^{5} + \frac{1333}{3564} a^{4} - \frac{1}{6} a^{3} - \frac{7}{33} a^{2} - \frac{5}{44} a - \frac{5}{22}$, $\frac{1}{233844319832082192684} a^{17} + \frac{56142442563107}{4330450367260781346} a^{16} + \frac{844711136549959}{77948106610694064228} a^{15} - \frac{87102131637034160}{58461079958020548171} a^{14} + \frac{74322228738843409}{8660900734521562692} a^{13} + \frac{62909138647517959}{12991351101782344038} a^{12} + \frac{326533284683480945}{116922159916041096342} a^{11} - \frac{41418927467696401}{4330450367260781346} a^{10} - \frac{2746552539276380789}{38974053305347032114} a^{9} - \frac{3048049056233155981}{38974053305347032114} a^{8} + \frac{365758782099626731}{6495675550891172019} a^{7} + \frac{24914228021602003}{316862221994691318} a^{6} - \frac{2939060293254360809}{25982702203564688076} a^{5} - \frac{40897731007391531}{1443483455753593782} a^{4} + \frac{249389383427993987}{2886966911507187564} a^{3} - \frac{34209198159723017}{80193525319644099} a^{2} + \frac{42760152078247387}{320774101278576396} a + \frac{21683325530614403}{53462350213096066}$

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:  $17$
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:  \( 942106579739 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T217:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 14 conjugacy class representatives for t18n217
Character table for t18n217

Intermediate fields

\(\Q(\sqrt{141}) \), 3.3.564.1 x3, 6.6.44851536.2, 9.9.165968803220544.1

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

Sibling fields

Degree 9 sibling: data not computed
Degree 12 siblings: data not computed
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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$$[\ ]_{3}^{2}$
$3$3.6.9.16$x^{6} + 3 x^{4} + 6 x^{3} + 3$$6$$1$$9$$S_3^2$$[3/2, 2]_{2}^{2}$
3.6.9.16$x^{6} + 3 x^{4} + 6 x^{3} + 3$$6$$1$$9$$S_3^2$$[3/2, 2]_{2}^{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
47Data not computed