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Properties

Label 18.0.23325000591...2831.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 13^{9}\cdot 19^{16}$
Root discriminant $256.64$
Ramified primes $3, 13, 19$
Class number $25138596$ (GRH)
Class group $[19, 1323084]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![427034554111, -104866576881, 68468181699, -8669431748, 7933667079, -1067345487, 527869077, -34322004, 33730047, -4191132, 1331088, -22440, 49286, -9846, 1686, 32, 12, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 12*x^16 + 32*x^15 + 1686*x^14 - 9846*x^13 + 49286*x^12 - 22440*x^11 + 1331088*x^10 - 4191132*x^9 + 33730047*x^8 - 34322004*x^7 + 527869077*x^6 - 1067345487*x^5 + 7933667079*x^4 - 8669431748*x^3 + 68468181699*x^2 - 104866576881*x + 427034554111)
 
gp: K = bnfinit(x^18 - 9*x^17 + 12*x^16 + 32*x^15 + 1686*x^14 - 9846*x^13 + 49286*x^12 - 22440*x^11 + 1331088*x^10 - 4191132*x^9 + 33730047*x^8 - 34322004*x^7 + 527869077*x^6 - 1067345487*x^5 + 7933667079*x^4 - 8669431748*x^3 + 68468181699*x^2 - 104866576881*x + 427034554111, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 12 x^{16} + 32 x^{15} + 1686 x^{14} - 9846 x^{13} + 49286 x^{12} - 22440 x^{11} + 1331088 x^{10} - 4191132 x^{9} + 33730047 x^{8} - 34322004 x^{7} + 527869077 x^{6} - 1067345487 x^{5} + 7933667079 x^{4} - 8669431748 x^{3} + 68468181699 x^{2} - 104866576881 x + 427034554111 \)

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:  \(-23325000591813882581633268226911166306572831=-\,3^{27}\cdot 13^{9}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $256.64$
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:  $3, 13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2223=3^{2}\cdot 13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{2223}(1600,·)$, $\chi_{2223}(1,·)$, $\chi_{2223}(196,·)$, $\chi_{2223}(389,·)$, $\chi_{2223}(2183,·)$, $\chi_{2223}(1676,·)$, $\chi_{2223}(272,·)$, $\chi_{2223}(1873,·)$, $\chi_{2223}(467,·)$, $\chi_{2223}(662,·)$, $\chi_{2223}(1052,·)$, $\chi_{2223}(157,·)$, $\chi_{2223}(235,·)$, $\chi_{2223}(1327,·)$, $\chi_{2223}(625,·)$, $\chi_{2223}(818,·)$, $\chi_{2223}(1715,·)$, $\chi_{2223}(313,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{33} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{4}{11} a$, $\frac{1}{33} a^{12} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{4}{11} a^{2} + \frac{1}{3}$, $\frac{1}{33} a^{13} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{4}{11} a^{3} + \frac{1}{3} a$, $\frac{1}{33} a^{14} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{4}{11} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{99} a^{15} - \frac{1}{99} a^{14} - \frac{1}{99} a^{13} + \frac{1}{99} a^{12} - \frac{1}{99} a^{11} - \frac{1}{9} a^{8} + \frac{4}{9} a^{7} + \frac{1}{9} a^{6} + \frac{32}{99} a^{5} + \frac{34}{99} a^{4} + \frac{23}{99} a^{3} - \frac{5}{11} a^{2} - \frac{32}{99} a - \frac{4}{9}$, $\frac{1}{36543615471} a^{16} - \frac{170838077}{36543615471} a^{15} + \frac{548522}{369127429} a^{14} + \frac{499544246}{36543615471} a^{13} + \frac{360668725}{36543615471} a^{12} - \frac{37243498}{3322146861} a^{11} + \frac{12106371}{369127429} a^{10} + \frac{510340643}{3322146861} a^{9} - \frac{99345881}{302013351} a^{8} - \frac{27802715}{100671117} a^{7} - \frac{1369196746}{4060401719} a^{6} - \frac{17074787551}{36543615471} a^{5} - \frac{1081319005}{3322146861} a^{4} + \frac{8508910396}{36543615471} a^{3} + \frac{6230768116}{36543615471} a^{2} - \frac{384359738}{1107382287} a - \frac{23316835}{302013351}$, $\frac{1}{2822190213605192588704621818420776255028072733849957816229693739} a^{17} + \frac{26362975743979181493731481553764237363510318874504559}{2822190213605192588704621818420776255028072733849957816229693739} a^{16} + \frac{550291795975491778104063440693282284612620481266868233142151}{940730071201730862901540606140258751676024244616652605409897913} a^{15} - \frac{11059152897189836352755279996661132213599892425606191097349983}{940730071201730862901540606140258751676024244616652605409897913} a^{14} - \frac{595389895354726833877979548322371699304350090261911752150872}{256562746691381144427692892583706932275279339440905256020881249} a^{13} + \frac{20671164979035161258603460521509825620792841983938700543565087}{2822190213605192588704621818420776255028072733849957816229693739} a^{12} + \frac{618700299383649374616017955795172224780097581126914582675386}{256562746691381144427692892583706932275279339440905256020881249} a^{11} + \frac{35554478556508516306590754422326213206001279951121987864207410}{256562746691381144427692892583706932275279339440905256020881249} a^{10} - \frac{6440927251260394212046061377740984434474806273435103587288046}{256562746691381144427692892583706932275279339440905256020881249} a^{9} - \frac{710420796936402476494460236167503121222279607399059773388887}{2591542895872536812400938308926332649245245852938436929503851} a^{8} + \frac{606180189453317707143670897912567406949576710879848298125485247}{2822190213605192588704621818420776255028072733849957816229693739} a^{7} + \frac{58374701628446014035928951018721690142023371396513265747023001}{940730071201730862901540606140258751676024244616652605409897913} a^{6} + \frac{907636160369144736514141140830568433397482889243515122214146010}{2822190213605192588704621818420776255028072733849957816229693739} a^{5} - \frac{541194059749128803748083375568626672151564343547264592013063589}{2822190213605192588704621818420776255028072733849957816229693739} a^{4} + \frac{11010643061583654076012447364630287696457556079249386644534045}{28506971854597904936410321398189659141697704382322806224542361} a^{3} - \frac{35318154735639953039114303547422321295057084804960205815259145}{2822190213605192588704621818420776255028072733849957816229693739} a^{2} + \frac{63857567780644415113871920283663082782283444704196244733554544}{256562746691381144427692892583706932275279339440905256020881249} a - \frac{10290728782441140086297258317969743430530668780722366051342664}{23323886062852831311608444780336993843207212676445932365534659}$

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

Class group and class number

$C_{19}\times C_{1323084}$, which has order $25138596$ (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:  \( 15010229.973756868 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

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

Intermediate fields

\(\Q(\sqrt{-39}) \), 3.3.361.1, 6.0.7730511399.2, 9.9.9025761726072081.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{18}$ R $18$ R $18$ $18$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
3Data not computed
13Data not computed
19Data not computed