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Properties

Label 18.0.15900874354...7751.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{15}\cdot 17^{9}$
Root discriminant $90.29$
Ramified primes $3, 7, 17$
Class number $294840$ (GRH)
Class group $[2, 18, 8190]$ (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![2115047096, -999364212, 1263575190, -534000599, 432944661, -156514464, 87289860, -25819998, 10996740, -2591186, 946242, -170670, 58932, -7524, 2574, -208, 72, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 72*x^16 - 208*x^15 + 2574*x^14 - 7524*x^13 + 58932*x^12 - 170670*x^11 + 946242*x^10 - 2591186*x^9 + 10996740*x^8 - 25819998*x^7 + 87289860*x^6 - 156514464*x^5 + 432944661*x^4 - 534000599*x^3 + 1263575190*x^2 - 999364212*x + 2115047096)
 
gp: K = bnfinit(x^18 - 3*x^17 + 72*x^16 - 208*x^15 + 2574*x^14 - 7524*x^13 + 58932*x^12 - 170670*x^11 + 946242*x^10 - 2591186*x^9 + 10996740*x^8 - 25819998*x^7 + 87289860*x^6 - 156514464*x^5 + 432944661*x^4 - 534000599*x^3 + 1263575190*x^2 - 999364212*x + 2115047096, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 72 x^{16} - 208 x^{15} + 2574 x^{14} - 7524 x^{13} + 58932 x^{12} - 170670 x^{11} + 946242 x^{10} - 2591186 x^{9} + 10996740 x^{8} - 25819998 x^{7} + 87289860 x^{6} - 156514464 x^{5} + 432944661 x^{4} - 534000599 x^{3} + 1263575190 x^{2} - 999364212 x + 2115047096 \)

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:  \(-159008743546498260206901265688807751=-\,3^{24}\cdot 7^{15}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $90.29$
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:  $3, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1071=3^{2}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1071}(256,·)$, $\chi_{1071}(1,·)$, $\chi_{1071}(358,·)$, $\chi_{1071}(577,·)$, $\chi_{1071}(970,·)$, $\chi_{1071}(715,·)$, $\chi_{1071}(205,·)$, $\chi_{1071}(271,·)$, $\chi_{1071}(919,·)$, $\chi_{1071}(985,·)$, $\chi_{1071}(475,·)$, $\chi_{1071}(220,·)$, $\chi_{1071}(832,·)$, $\chi_{1071}(613,·)$, $\chi_{1071}(934,·)$, $\chi_{1071}(562,·)$, $\chi_{1071}(628,·)$, $\chi_{1071}(118,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{362} a^{12} + \frac{67}{362} a^{11} + \frac{12}{181} a^{10} + \frac{81}{362} a^{9} - \frac{42}{181} a^{8} - \frac{79}{181} a^{7} + \frac{41}{181} a^{6} - \frac{39}{362} a^{5} + \frac{7}{362} a^{4} + \frac{4}{181} a^{3} + \frac{69}{362} a^{2} + \frac{1}{181} a - \frac{69}{181}$, $\frac{1}{362} a^{13} + \frac{30}{181} a^{11} - \frac{79}{362} a^{10} - \frac{81}{362} a^{9} + \frac{20}{181} a^{8} + \frac{85}{181} a^{7} - \frac{103}{362} a^{6} + \frac{43}{181} a^{5} + \frac{41}{181} a^{4} - \frac{105}{362} a^{3} + \frac{85}{362} a^{2} + \frac{45}{181} a - \frac{83}{181}$, $\frac{1}{362} a^{14} + \frac{32}{181} a^{11} - \frac{73}{362} a^{10} + \frac{67}{362} a^{9} - \frac{39}{362} a^{8} - \frac{35}{362} a^{7} - \frac{64}{181} a^{6} - \frac{56}{181} a^{5} + \frac{9}{181} a^{4} - \frac{33}{362} a^{3} + \frac{113}{362} a^{2} - \frac{105}{362} a - \frac{23}{181}$, $\frac{1}{724} a^{15} + \frac{41}{181} a^{11} + \frac{40}{181} a^{10} + \frac{13}{362} a^{9} + \frac{23}{181} a^{8} + \frac{105}{362} a^{7} - \frac{73}{181} a^{6} - \frac{5}{181} a^{5} - \frac{75}{181} a^{4} + \frac{36}{181} a^{3} - \frac{179}{362} a^{2} + \frac{7}{724} a - \frac{109}{362}$, $\frac{1}{1448} a^{16} - \frac{1}{1448} a^{15} - \frac{1}{724} a^{12} + \frac{12}{181} a^{11} - \frac{17}{181} a^{10} + \frac{7}{724} a^{9} + \frac{153}{724} a^{8} + \frac{193}{724} a^{7} + \frac{52}{181} a^{6} + \frac{201}{724} a^{5} + \frac{46}{181} a^{4} - \frac{5}{362} a^{3} - \frac{591}{1448} a^{2} + \frac{167}{1448} a + \frac{341}{724}$, $\frac{1}{3613798383603951106090191271374998685435838560177526635968} a^{17} + \frac{80921885912643986650694837841876801135460437744345235}{3613798383603951106090191271374998685435838560177526635968} a^{16} + \frac{526498013654398211106253359970370410526961068919440133}{1806899191801975553045095635687499342717919280088763317984} a^{15} - \frac{411485169363401067237140437884857991597870957221938541}{903449595900987776522547817843749671358959640044381658992} a^{14} - \frac{935569246780975466942209715660778722367638853666737397}{1806899191801975553045095635687499342717919280088763317984} a^{13} - \frac{17058621082084308872082958184235484854055823181397983}{56465599743811736032659238615234354459934977502773853687} a^{12} - \frac{205472366517912633435717318369209511527855049784283058147}{903449595900987776522547817843749671358959640044381658992} a^{11} + \frac{108333299493923074239948162757319843832731454332818201685}{1806899191801975553045095635687499342717919280088763317984} a^{10} + \frac{156402388448638864313474509751002248561209831147480144127}{1806899191801975553045095635687499342717919280088763317984} a^{9} - \frac{9386412818210302257208901248810961209602925936507635295}{1806899191801975553045095635687499342717919280088763317984} a^{8} - \frac{89542557964341351746602846675343887403205503173452521881}{225862398975246944130636954460937417839739910011095414748} a^{7} - \frac{481581140164379007130802515366309166845619113259145107935}{1806899191801975553045095635687499342717919280088763317984} a^{6} + \frac{35129155618976270349562437310427013379274960198543008715}{225862398975246944130636954460937417839739910011095414748} a^{5} + \frac{375273258650450246576266421471466902946649777689230356}{56465599743811736032659238615234354459934977502773853687} a^{4} - \frac{1792690523411715065180157803688351532592787051286571157131}{3613798383603951106090191271374998685435838560177526635968} a^{3} - \frac{876494928393415515262149822901878087302731088988716810793}{3613798383603951106090191271374998685435838560177526635968} a^{2} + \frac{71423280615921263985295288421553468331272118598191244155}{225862398975246944130636954460937417839739910011095414748} a - \frac{240403527554019729011108107447632745815756609226077352549}{903449595900987776522547817843749671358959640044381658992}$

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

Class group and class number

$C_{2}\times C_{18}\times C_{8190}$, which has order $294840$ (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:  \( 54408.4888887 \) (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

\(\Q(\sqrt{-119}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.1, 3.3.3969.2, \(\Q(\zeta_{7})^+\), 6.0.11056328199.5, 6.0.541760081751.3, 6.0.541760081751.4, 6.0.82572791.1, 9.9.62523502209.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
7Data not computed
$17$17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$