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Properties

Label 18.0.21005238210...5344.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 7^{15}\cdot 19^{12}$
Root discriminant $374.51$
Ramified primes $2, 3, 7, 19$
Class number $188225856$ (GRH)
Class group $[2, 6, 36, 435708]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![250641775000, -148747023000, 347083985700, -82064768620, 109092265428, 13683356496, 4501586641, 37355694, 216555534, 19740598, 2658099, -1038168, -7532, 2706, 2034, -456, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 30*x^16 - 456*x^15 + 2034*x^14 + 2706*x^13 - 7532*x^12 - 1038168*x^11 + 2658099*x^10 + 19740598*x^9 + 216555534*x^8 + 37355694*x^7 + 4501586641*x^6 + 13683356496*x^5 + 109092265428*x^4 - 82064768620*x^3 + 347083985700*x^2 - 148747023000*x + 250641775000)
 
gp: K = bnfinit(x^18 - 30*x^16 - 456*x^15 + 2034*x^14 + 2706*x^13 - 7532*x^12 - 1038168*x^11 + 2658099*x^10 + 19740598*x^9 + 216555534*x^8 + 37355694*x^7 + 4501586641*x^6 + 13683356496*x^5 + 109092265428*x^4 - 82064768620*x^3 + 347083985700*x^2 - 148747023000*x + 250641775000, 1)
 

Normalized defining polynomial

\( x^{18} - 30 x^{16} - 456 x^{15} + 2034 x^{14} + 2706 x^{13} - 7532 x^{12} - 1038168 x^{11} + 2658099 x^{10} + 19740598 x^{9} + 216555534 x^{8} + 37355694 x^{7} + 4501586641 x^{6} + 13683356496 x^{5} + 109092265428 x^{4} - 82064768620 x^{3} + 347083985700 x^{2} - 148747023000 x + 250641775000 \)

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:  \(-21005238210476463669563930923405113491534905344=-\,2^{18}\cdot 3^{27}\cdot 7^{15}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $374.51$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4788=2^{2}\cdot 3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4788}(1,·)$, $\chi_{4788}(1223,·)$, $\chi_{4788}(457,·)$, $\chi_{4788}(3659,·)$, $\chi_{4788}(1679,·)$, $\chi_{4788}(1873,·)$, $\chi_{4788}(4115,·)$, $\chi_{4788}(2965,·)$, $\chi_{4788}(1559,·)$, $\chi_{4788}(2015,·)$, $\chi_{4788}(3697,·)$, $\chi_{4788}(2857,·)$, $\chi_{4788}(3503,·)$, $\chi_{4788}(3313,·)$, $\chi_{4788}(1033,·)$, $\chi_{4788}(4153,·)$, $\chi_{4788}(1151,·)$, $\chi_{4788}(3839,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{5} a^{6} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{7} - \frac{1}{5} a^{3}$, $\frac{1}{10} a^{8} - \frac{1}{10} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{9} - \frac{1}{10} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{50} a^{10} + \frac{3}{50} a^{6} - \frac{1}{2} a^{4} - \frac{2}{25} a^{2}$, $\frac{1}{50} a^{11} + \frac{3}{50} a^{7} - \frac{1}{10} a^{5} - \frac{2}{25} a^{3} - \frac{2}{5} a$, $\frac{1}{550} a^{12} - \frac{2}{275} a^{11} - \frac{1}{55} a^{9} - \frac{1}{25} a^{8} - \frac{1}{275} a^{7} - \frac{1}{22} a^{6} + \frac{1}{11} a^{5} + \frac{271}{550} a^{4} + \frac{128}{275} a^{3} + \frac{1}{22} a^{2} - \frac{24}{55} a + \frac{3}{11}$, $\frac{1}{2750} a^{13} - \frac{1}{2750} a^{12} - \frac{6}{1375} a^{11} - \frac{1}{275} a^{10} + \frac{29}{1375} a^{9} + \frac{21}{1375} a^{8} + \frac{79}{2750} a^{7} + \frac{39}{550} a^{6} + \frac{91}{2750} a^{5} + \frac{409}{2750} a^{4} - \frac{417}{2750} a^{3} - \frac{17}{50} a^{2} - \frac{7}{55} a + \frac{4}{11}$, $\frac{1}{277750} a^{14} + \frac{19}{277750} a^{13} - \frac{177}{277750} a^{12} - \frac{7}{1010} a^{11} - \frac{967}{277750} a^{10} - \frac{13573}{277750} a^{9} - \frac{5233}{138875} a^{8} - \frac{446}{5555} a^{7} + \frac{6791}{277750} a^{6} + \frac{13129}{277750} a^{5} - \frac{22916}{138875} a^{4} - \frac{1523}{5555} a^{3} + \frac{1642}{5555} a^{2} - \frac{382}{1111} a + \frac{455}{1111}$, $\frac{1}{518003750} a^{15} + \frac{631}{518003750} a^{14} - \frac{42906}{259001875} a^{13} + \frac{201567}{259001875} a^{12} + \frac{3589749}{518003750} a^{11} + \frac{1486274}{259001875} a^{10} + \frac{365189}{47091250} a^{9} + \frac{7025836}{259001875} a^{8} - \frac{23954903}{259001875} a^{7} + \frac{15445871}{518003750} a^{6} + \frac{15485429}{259001875} a^{5} - \frac{97959428}{259001875} a^{4} + \frac{87675328}{259001875} a^{3} + \frac{292663}{4144030} a^{2} + \frac{1080908}{10360075} a + \frac{17817}{414403}$, $\frac{1}{1036007500} a^{16} - \frac{469}{518003750} a^{14} - \frac{31891}{259001875} a^{13} - \frac{12092}{51800375} a^{12} + \frac{3971587}{518003750} a^{11} - \frac{286476}{259001875} a^{10} + \frac{9411597}{259001875} a^{9} - \frac{3259493}{1036007500} a^{8} + \frac{1687271}{518003750} a^{7} - \frac{20382902}{259001875} a^{6} - \frac{1911937}{518003750} a^{5} - \frac{240197393}{1036007500} a^{4} - \frac{80151629}{259001875} a^{3} - \frac{7506839}{20720150} a^{2} - \frac{286549}{941825} a - \frac{106272}{414403}$, $\frac{1}{17140560628872487095435395607677483019648084342574126685087500} a^{17} - \frac{1567094952914358116314493831855074301767160165755039}{4285140157218121773858848901919370754912021085643531671271875} a^{16} + \frac{2327497034740560534204455034254662291143040911299609}{4285140157218121773858848901919370754912021085643531671271875} a^{15} + \frac{173889550535704585945189591468451976569364817693847562}{4285140157218121773858848901919370754912021085643531671271875} a^{14} - \frac{438181220006421130575508545058121953113607810484123067606}{4285140157218121773858848901919370754912021085643531671271875} a^{13} + \frac{746285645131339527536705226314017160867419788189023111602}{857028031443624354771769780383874150982404217128706334254375} a^{12} - \frac{34187617961232250830404002586566000311241253547831174741798}{4285140157218121773858848901919370754912021085643531671271875} a^{11} - \frac{18042454394184090831773620756468212227937267883932639855283}{8570280314436243547717697803838741509824042171287063342543750} a^{10} - \frac{124306228643300069224280131859400109010099095553453159934203}{3428112125774497419087079121535496603929616868514825337017500} a^{9} - \frac{201594955593375431578686047689228765840983289571277186907073}{4285140157218121773858848901919370754912021085643531671271875} a^{8} + \frac{823728866393547903991386142835567504047096951887873581245493}{8570280314436243547717697803838741509824042171287063342543750} a^{7} + \frac{229920689934313590387309868648685531964914753468513748887009}{8570280314436243547717697803838741509824042171287063342543750} a^{6} - \frac{39920473468346903064319886251466750248123485202511849422487}{17140560628872487095435395607677483019648084342574126685087500} a^{5} - \frac{307963032500163397789523427867910454606576891491449376254511}{779116392221476686156154345803521955438549288298823940231250} a^{4} - \frac{87477612449181897228588070590324422407872553404146571927143}{1714056062887248709543539560767748301964808434257412668508750} a^{3} - \frac{82483604826476818942409977880355483602566971744178684093233}{171405606288724870954353956076774830196480843425741266850875} a^{2} - \frac{7198623214456060099996999776231699652382214849009737890389}{34281121257744974190870791215354966039296168685148253370175} a - \frac{194752920236976663351625643516659564026645903505418155482}{1371244850309798967634831648614198641571846747405930134807}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{36}\times C_{435708}$, which has order $188225856$ (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:  \( 926168257.1203558 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-21}) \), 3.3.1432809.4, 3.3.3969.2, 3.3.17689.2, 3.3.29241.2, 6.0.2759153551366464.3, 6.0.21171979584.1, 6.0.3784847121216.2, 6.0.56309256150336.2, 9.9.2941473244627851129.6

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

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 ${\href{/LocalNumberField/5.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
7Data not computed
$19$19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$