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Properties

Label 18.0.29585973151...2443.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 521^{9}$
Root discriminant $82.24$
Ramified primes $3, 521$
Class number $14450$ (GRH)
Class group $[85, 170]$ (GRH)
Galois group $D_9$ (as 18T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![207974400, 180725760, 41054208, 8265216, 24462336, 14185344, 1444480, 635424, 654144, -5656, 7536, 16026, 3951, -1161, 1116, -157, 60, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 60*x^16 - 157*x^15 + 1116*x^14 - 1161*x^13 + 3951*x^12 + 16026*x^11 + 7536*x^10 - 5656*x^9 + 654144*x^8 + 635424*x^7 + 1444480*x^6 + 14185344*x^5 + 24462336*x^4 + 8265216*x^3 + 41054208*x^2 + 180725760*x + 207974400)
 
gp: K = bnfinit(x^18 - 3*x^17 + 60*x^16 - 157*x^15 + 1116*x^14 - 1161*x^13 + 3951*x^12 + 16026*x^11 + 7536*x^10 - 5656*x^9 + 654144*x^8 + 635424*x^7 + 1444480*x^6 + 14185344*x^5 + 24462336*x^4 + 8265216*x^3 + 41054208*x^2 + 180725760*x + 207974400, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 60 x^{16} - 157 x^{15} + 1116 x^{14} - 1161 x^{13} + 3951 x^{12} + 16026 x^{11} + 7536 x^{10} - 5656 x^{9} + 654144 x^{8} + 635424 x^{7} + 1444480 x^{6} + 14185344 x^{5} + 24462336 x^{4} + 8265216 x^{3} + 41054208 x^{2} + 180725760 x + 207974400 \)

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:  \(-29585973151813465955360437434322443=-\,3^{21}\cdot 521^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.24$
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:  $3, 521$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} + \frac{3}{16} a^{4} + \frac{1}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{64} a^{10} + \frac{1}{64} a^{9} + \frac{3}{64} a^{7} - \frac{1}{8} a^{6} - \frac{9}{64} a^{5} + \frac{11}{64} a^{4} + \frac{3}{32} a^{3} + \frac{1}{8} a^{2} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{128} a^{11} - \frac{1}{128} a^{10} - \frac{1}{64} a^{9} - \frac{13}{128} a^{8} + \frac{1}{64} a^{7} - \frac{9}{128} a^{6} - \frac{19}{128} a^{5} - \frac{1}{4} a^{4} + \frac{3}{32} a^{3} - \frac{5}{16} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{15360} a^{12} + \frac{13}{5120} a^{11} + \frac{7}{2560} a^{10} + \frac{199}{15360} a^{9} + \frac{231}{2560} a^{8} - \frac{479}{5120} a^{7} - \frac{299}{15360} a^{6} + \frac{167}{1280} a^{5} + \frac{73}{640} a^{4} - \frac{3}{80} a^{3} + \frac{31}{160} a^{2} + \frac{3}{10} a - \frac{7}{16}$, $\frac{1}{30720} a^{13} - \frac{1}{30720} a^{12} - \frac{13}{5120} a^{11} - \frac{41}{30720} a^{10} - \frac{407}{15360} a^{9} + \frac{1041}{10240} a^{8} - \frac{419}{30720} a^{7} + \frac{731}{7680} a^{6} - \frac{47}{1280} a^{5} + \frac{7}{160} a^{4} + \frac{11}{320} a^{3} - \frac{1}{10} a^{2} - \frac{15}{32} a + \frac{1}{4}$, $\frac{1}{14008320} a^{14} + \frac{217}{14008320} a^{13} + \frac{1}{184320} a^{12} - \frac{51497}{14008320} a^{11} + \frac{1627}{875520} a^{10} - \frac{119501}{14008320} a^{9} - \frac{1105133}{14008320} a^{8} + \frac{679229}{7004160} a^{7} - \frac{151211}{3502080} a^{6} + \frac{4241}{18240} a^{5} + \frac{1867}{9120} a^{4} + \frac{7447}{72960} a^{3} - \frac{36317}{72960} a^{2} + \frac{3229}{36480} a - \frac{1727}{3648}$, $\frac{1}{140083200} a^{15} - \frac{1}{28016640} a^{14} - \frac{793}{70041600} a^{13} - \frac{3617}{140083200} a^{12} + \frac{175103}{70041600} a^{11} - \frac{505949}{140083200} a^{10} - \frac{363623}{140083200} a^{9} + \frac{400549}{8755200} a^{8} + \frac{1693643}{17510400} a^{7} - \frac{2693}{5836800} a^{6} - \frac{14653}{182400} a^{5} + \frac{22963}{729600} a^{4} - \frac{2807}{729600} a^{3} - \frac{4423}{45600} a^{2} - \frac{2893}{18240} a - \frac{277}{1216}$, $\frac{1}{3279067545600} a^{16} - \frac{5821}{3279067545600} a^{15} + \frac{32957}{1639533772800} a^{14} - \frac{53304701}{3279067545600} a^{13} + \frac{4513877}{234219110400} a^{12} - \frac{1032320173}{655813509120} a^{11} - \frac{13185357839}{3279067545600} a^{10} - \frac{2695657787}{819766886400} a^{9} - \frac{9196097599}{163953377280} a^{8} + \frac{1511548581}{22771302400} a^{7} - \frac{10143424087}{204941721600} a^{6} - \frac{473458021}{3415695360} a^{5} + \frac{128395719}{1138565120} a^{4} - \frac{826779439}{4269619200} a^{3} - \frac{143889961}{1423206400} a^{2} + \frac{885023}{14232064} a - \frac{16638719}{42696192}$, $\frac{1}{8158530733631303577108480000} a^{17} + \frac{20072695316577}{181300682969584523935744000} a^{16} - \frac{62602383669015811}{67987756113594196475904000} a^{15} - \frac{82083257525217671417}{8158530733631303577108480000} a^{14} - \frac{782372900605113310261}{58275219525937882693632000} a^{13} + \frac{79490324577131177238619}{8158530733631303577108480000} a^{12} + \frac{21920156422624275342071143}{8158530733631303577108480000} a^{11} + \frac{330259474122986992652759}{163170614672626071542169600} a^{10} - \frac{768723042214065721706029}{70332161496821582561280000} a^{9} - \frac{3920763902264540751698389}{113312926855990327459840000} a^{8} + \frac{1352029291651133873827891}{20396326834078258942771200} a^{7} - \frac{27010838466818225467602793}{254954085425978236784640000} a^{6} + \frac{7893961634042025456593711}{42492347570996372797440000} a^{5} + \frac{197172952563202380453921}{1416411585699879093248000} a^{4} - \frac{1526422495694456643889313}{10623086892749093199360000} a^{3} + \frac{102463413001396967613139}{5311543446374546599680000} a^{2} - \frac{4723604485903891869493}{18315667056463953792000} a - \frac{5383861996471744777}{11208152450674291200}$

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

Class group and class number

$C_{85}\times C_{170}$, which has order $14450$ (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:  \( 14940330.1709 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9$ (as 18T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $D_9$
Character table for $D_9$

Intermediate fields

\(\Q(\sqrt{-1563}) \), 3.1.1563.1 x3, 6.0.3818360547.1, 9.1.4350743102986569.1 x9

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
521Data not computed