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Properties

Label 18.0.29091207730...7531.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,11^{9}\cdot 37^{16}$
Root discriminant $82.16$
Ramified primes $11, 37$
Class number $12996$ (GRH)
Class group $[38, 342]$ (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![4399861, -3731965, 5252508, -2871539, 2318748, -1068295, 710103, -309033, 153844, -51876, 19848, -6459, 2399, -547, 41, 6, 15, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 15*x^16 + 6*x^15 + 41*x^14 - 547*x^13 + 2399*x^12 - 6459*x^11 + 19848*x^10 - 51876*x^9 + 153844*x^8 - 309033*x^7 + 710103*x^6 - 1068295*x^5 + 2318748*x^4 - 2871539*x^3 + 5252508*x^2 - 3731965*x + 4399861)
 
gp: K = bnfinit(x^18 - 7*x^17 + 15*x^16 + 6*x^15 + 41*x^14 - 547*x^13 + 2399*x^12 - 6459*x^11 + 19848*x^10 - 51876*x^9 + 153844*x^8 - 309033*x^7 + 710103*x^6 - 1068295*x^5 + 2318748*x^4 - 2871539*x^3 + 5252508*x^2 - 3731965*x + 4399861, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 15 x^{16} + 6 x^{15} + 41 x^{14} - 547 x^{13} + 2399 x^{12} - 6459 x^{11} + 19848 x^{10} - 51876 x^{9} + 153844 x^{8} - 309033 x^{7} + 710103 x^{6} - 1068295 x^{5} + 2318748 x^{4} - 2871539 x^{3} + 5252508 x^{2} - 3731965 x + 4399861 \)

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:  \(-29091207730813357496487416074727531=-\,11^{9}\cdot 37^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.16$
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:  $11, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(407=11\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{407}(1,·)$, $\chi_{407}(386,·)$, $\chi_{407}(197,·)$, $\chi_{407}(329,·)$, $\chi_{407}(10,·)$, $\chi_{407}(12,·)$, $\chi_{407}(144,·)$, $\chi_{407}(340,·)$, $\chi_{407}(342,·)$, $\chi_{407}(100,·)$, $\chi_{407}(155,·)$, $\chi_{407}(285,·)$, $\chi_{407}(34,·)$, $\chi_{407}(219,·)$, $\chi_{407}(164,·)$, $\chi_{407}(232,·)$, $\chi_{407}(120,·)$, $\chi_{407}(186,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{31} a^{15} + \frac{4}{31} a^{14} - \frac{7}{31} a^{13} + \frac{8}{31} a^{12} + \frac{10}{31} a^{11} + \frac{13}{31} a^{10} + \frac{7}{31} a^{9} + \frac{3}{31} a^{8} + \frac{8}{31} a^{7} + \frac{15}{31} a^{6} + \frac{6}{31} a^{5} - \frac{3}{31} a^{4} - \frac{10}{31} a^{3} + \frac{3}{31} a^{2} + \frac{15}{31} a$, $\frac{1}{1333} a^{16} + \frac{11}{1333} a^{15} + \frac{269}{1333} a^{14} - \frac{599}{1333} a^{13} + \frac{66}{1333} a^{12} + \frac{207}{1333} a^{11} - \frac{26}{1333} a^{10} + \frac{21}{1333} a^{9} - \frac{64}{1333} a^{8} + \frac{350}{1333} a^{7} - \frac{261}{1333} a^{6} - \frac{54}{1333} a^{5} - \frac{9}{43} a^{4} + \frac{336}{1333} a^{3} - \frac{57}{1333} a^{2} - \frac{484}{1333} a - \frac{9}{43}$, $\frac{1}{26570325676916631668769417512268693507082739051} a^{17} - \frac{1924168803280280761081044057510712679622977}{26570325676916631668769417512268693507082739051} a^{16} + \frac{235863176984900442622086069713856887733724981}{26570325676916631668769417512268693507082739051} a^{15} - \frac{4518388921776526497532877762083318762674223453}{26570325676916631668769417512268693507082739051} a^{14} - \frac{13144968843490229682962573356192500077483716119}{26570325676916631668769417512268693507082739051} a^{13} + \frac{3659950443341386142510423502079699862349512428}{26570325676916631668769417512268693507082739051} a^{12} + \frac{10838249468852161389847074427857189121393971894}{26570325676916631668769417512268693507082739051} a^{11} - \frac{11469231709694960600194850977337146488849926886}{26570325676916631668769417512268693507082739051} a^{10} + \frac{3804783809920511602092651788346164824085942540}{26570325676916631668769417512268693507082739051} a^{9} + \frac{8220671950369623632889350380937357666655921309}{26570325676916631668769417512268693507082739051} a^{8} - \frac{1583685815565626600805661261829605371000910849}{26570325676916631668769417512268693507082739051} a^{7} + \frac{5367903185939226022547496775788792408040779690}{26570325676916631668769417512268693507082739051} a^{6} + \frac{12574491683064395310237858218290116873841283929}{26570325676916631668769417512268693507082739051} a^{5} - \frac{11335800913258336890973497901301964841773025330}{26570325676916631668769417512268693507082739051} a^{4} - \frac{286951434421776690927180308017202739245653499}{26570325676916631668769417512268693507082739051} a^{3} + \frac{5621340637056493092884443632167565799287688041}{26570325676916631668769417512268693507082739051} a^{2} + \frac{5702081261928695698670366209261627220643281035}{26570325676916631668769417512268693507082739051} a - \frac{186835558682143746543822834913363418390665558}{857107279900536505444174758460280435712346421}$

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

Class group and class number

$C_{38}\times C_{342}$, which has order $12996$ (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:  \( 409151.310213 \) (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

\(\Q(\sqrt{-11}) \), 3.3.1369.1, 6.0.2494508291.2, 9.9.3512479453921.1

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 $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ R $18$ $18$ $18$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ R $18$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$37.9.8.1$x^{9} - 37$$9$$1$$8$$C_9$$[\ ]_{9}$
37.9.8.1$x^{9} - 37$$9$$1$$8$$C_9$$[\ ]_{9}$