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Properties

Label 18.12.2781519119...1875.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,3^{24}\cdot 5^{9}\cdot 19^{2}\cdot 199\cdot 264935899^{2}$
Root discriminant $155.36$
Ramified primes $3, 5, 19, 199, 264935899$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T857

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![46470800799, 168126981534, 243718173702, 178677196446, 61730463267, -203134881, -7870564363, -2202823131, 118087020, 156699733, 16346880, -4347954, -900210, 46632, 19896, 39, -216, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 216*x^16 + 39*x^15 + 19896*x^14 + 46632*x^13 - 900210*x^12 - 4347954*x^11 + 16346880*x^10 + 156699733*x^9 + 118087020*x^8 - 2202823131*x^7 - 7870564363*x^6 - 203134881*x^5 + 61730463267*x^4 + 178677196446*x^3 + 243718173702*x^2 + 168126981534*x + 46470800799)
 
gp: K = bnfinit(x^18 - 3*x^17 - 216*x^16 + 39*x^15 + 19896*x^14 + 46632*x^13 - 900210*x^12 - 4347954*x^11 + 16346880*x^10 + 156699733*x^9 + 118087020*x^8 - 2202823131*x^7 - 7870564363*x^6 - 203134881*x^5 + 61730463267*x^4 + 178677196446*x^3 + 243718173702*x^2 + 168126981534*x + 46470800799, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 216 x^{16} + 39 x^{15} + 19896 x^{14} + 46632 x^{13} - 900210 x^{12} - 4347954 x^{11} + 16346880 x^{10} + 156699733 x^{9} + 118087020 x^{8} - 2202823131 x^{7} - 7870564363 x^{6} - 203134881 x^{5} + 61730463267 x^{4} + 178677196446 x^{3} + 243718173702 x^{2} + 168126981534 x + 46470800799 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2781519119947489262355092486938201171875=-\,3^{24}\cdot 5^{9}\cdot 19^{2}\cdot 199\cdot 264935899^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $155.36$
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:  $3, 5, 19, 199, 264935899$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{1686} a^{16} + \frac{9}{562} a^{15} + \frac{123}{562} a^{14} - \frac{4}{281} a^{13} + \frac{73}{562} a^{12} - \frac{68}{281} a^{11} + \frac{23}{281} a^{10} + \frac{13}{281} a^{9} - \frac{104}{281} a^{8} - \frac{250}{843} a^{7} + \frac{36}{281} a^{6} + \frac{9}{281} a^{5} + \frac{833}{1686} a^{4} - \frac{58}{281} a^{3} + \frac{84}{281} a^{2} - \frac{139}{562} a + \frac{70}{281}$, $\frac{1}{45721954964861417784204110091796742682848003187331881675797946} a^{17} - \frac{1150605404718527740630507160727778659899990768620026545791}{22860977482430708892102055045898371341424001593665940837898973} a^{16} - \frac{2615478043828022544073616637979590269083121957978351957225941}{45721954964861417784204110091796742682848003187331881675797946} a^{15} + \frac{3750924881751191690248999671852499868669449338483684658291013}{15240651654953805928068036697265580894282667729110627225265982} a^{14} - \frac{1054848744602944347222257766148032419137728450427253756518971}{15240651654953805928068036697265580894282667729110627225265982} a^{13} + \frac{3234129579355278338524561972835227830114090948015810348513581}{15240651654953805928068036697265580894282667729110627225265982} a^{12} + \frac{653479439321561708017360179476646252412780310097549921172937}{7620325827476902964034018348632790447141333864555313612632991} a^{11} + \frac{3629948783173212553263949349182120691991475671114346224384555}{15240651654953805928068036697265580894282667729110627225265982} a^{10} - \frac{3544757479038776017299841044838914451605115427723048881547087}{15240651654953805928068036697265580894282667729110627225265982} a^{9} + \frac{5241953914741396891323151262348095382136188450309340511419115}{45721954964861417784204110091796742682848003187331881675797946} a^{8} - \frac{15580418469777543343082322173829204570117837170700099804139499}{45721954964861417784204110091796742682848003187331881675797946} a^{7} - \frac{2616612644677252252102870837565373755198034888671430873916639}{45721954964861417784204110091796742682848003187331881675797946} a^{6} + \frac{10840082010180485143227187601534743673596396331097506922665983}{22860977482430708892102055045898371341424001593665940837898973} a^{5} - \frac{7685773351154342594294558408624159339164685645635526523132266}{22860977482430708892102055045898371341424001593665940837898973} a^{4} - \frac{4127567971079193908398955405003018883042504803532542047402942}{22860977482430708892102055045898371341424001593665940837898973} a^{3} - \frac{7346765365129821406902467432111594828003602774459745408623289}{15240651654953805928068036697265580894282667729110627225265982} a^{2} + \frac{3064797161647275849459029205503796881294876930497490188333891}{7620325827476902964034018348632790447141333864555313612632991} a + \frac{6772906926226893675720486557202281401439457966935315568883409}{15240651654953805928068036697265580894282667729110627225265982}$

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:  $14$
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:  \( 26834946087500 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T857:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 279936
The 159 conjugacy class representatives for t18n857 are not computed
Character table for t18n857 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 6.6.820125.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R R $18$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ $18$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$3$3.6.8.3$x^{6} + 18 x^{2} + 9$$3$$2$$8$$C_6$$[2]^{2}$
3.12.16.14$x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$$3$$4$$16$$C_{12}$$[2]^{4}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
199Data not computed
264935899Data not computed