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Properties

Label 18.0.72436308017...1552.5
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{44}\cdot 19^{17}$
Root discriminant $669.20$
Ramified primes $2, 3, 19$
Class number $2646942624$ (GRH)
Class group $[2, 36, 36763092]$ (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![1297774555430912, 0, 376568116936704, 0, 44181664137216, 0, 2718287880192, 0, 95935104000, 0, 2016760320, 0, 25307544, 0, 182628, 0, 684, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 684*x^16 + 182628*x^14 + 25307544*x^12 + 2016760320*x^10 + 95935104000*x^8 + 2718287880192*x^6 + 44181664137216*x^4 + 376568116936704*x^2 + 1297774555430912)
 
gp: K = bnfinit(x^18 + 684*x^16 + 182628*x^14 + 25307544*x^12 + 2016760320*x^10 + 95935104000*x^8 + 2718287880192*x^6 + 44181664137216*x^4 + 376568116936704*x^2 + 1297774555430912, 1)
 

Normalized defining polynomial

\( x^{18} + 684 x^{16} + 182628 x^{14} + 25307544 x^{12} + 2016760320 x^{10} + 95935104000 x^{8} + 2718287880192 x^{6} + 44181664137216 x^{4} + 376568116936704 x^{2} + 1297774555430912 \)

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:  \(-724363080171793764995208566309796540433577438871552=-\,2^{27}\cdot 3^{44}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $669.20$
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:  $2, 3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4104=2^{3}\cdot 3^{3}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4104}(1,·)$, $\chi_{4104}(901,·)$, $\chi_{4104}(1993,·)$, $\chi_{4104}(1873,·)$, $\chi_{4104}(2389,·)$, $\chi_{4104}(313,·)$, $\chi_{4104}(3481,·)$, $\chi_{4104}(829,·)$, $\chi_{4104}(925,·)$, $\chi_{4104}(2245,·)$, $\chi_{4104}(2353,·)$, $\chi_{4104}(2941,·)$, $\chi_{4104}(3313,·)$, $\chi_{4104}(2761,·)$, $\chi_{4104}(3577,·)$, $\chi_{4104}(637,·)$, $\chi_{4104}(1405,·)$, $\chi_{4104}(1237,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{16} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{1664} a^{9} + \frac{3}{416} a^{7} - \frac{3}{32} a^{5} - \frac{21}{208} a^{3} - \frac{6}{13} a$, $\frac{1}{6656} a^{10} + \frac{3}{1664} a^{8} - \frac{3}{128} a^{6} + \frac{83}{832} a^{4} - \frac{3}{26} a^{2}$, $\frac{1}{13312} a^{11} - \frac{1}{3328} a^{9} - \frac{87}{3328} a^{7} + \frac{187}{1664} a^{5} + \frac{15}{104} a^{3} - \frac{1}{13} a$, $\frac{1}{212992} a^{12} - \frac{1}{53248} a^{10} + \frac{121}{53248} a^{8} + \frac{1019}{26624} a^{6} + \frac{41}{1664} a^{4} - \frac{41}{416} a^{2} + \frac{1}{4}$, $\frac{1}{425984} a^{13} - \frac{1}{106496} a^{11} - \frac{7}{106496} a^{9} + \frac{251}{53248} a^{7} - \frac{167}{3328} a^{5} - \frac{81}{832} a^{3} + \frac{5}{104} a$, $\frac{1}{1703936} a^{14} - \frac{1}{425984} a^{12} - \frac{7}{425984} a^{10} + \frac{251}{212992} a^{8} - \frac{167}{13312} a^{6} - \frac{81}{3328} a^{4} + \frac{5}{416} a^{2}$, $\frac{1}{1496055808} a^{15} - \frac{5}{374013952} a^{13} + \frac{9}{374013952} a^{11} - \frac{28621}{187006976} a^{9} + \frac{117903}{5843968} a^{7} + \frac{101883}{2921984} a^{5} - \frac{1801}{28096} a^{3} + \frac{939}{22828} a$, $\frac{1}{1029446469540782137769743876096} a^{16} - \frac{4825030684703736800563}{257361617385195534442435969024} a^{14} + \frac{31226822032042108009365}{19797047491168887264802766848} a^{12} + \frac{4202734638361359928472343}{128680808692597767221217984512} a^{10} - \frac{117466289850373949433462525}{16085101086574720902652248064} a^{8} - \frac{477043405299178582387009}{38666108381189232939067904} a^{6} - \frac{523603036900149068142179}{62832426119432503525985344} a^{4} - \frac{1903259454516918486375331}{15708106529858125881496336} a^{2} - \frac{17435511272841724197}{52931307469430678524}$, $\frac{1}{2058892939081564275539487752192} a^{17} - \frac{8281704985545054879}{514723234770391068884871938048} a^{15} + \frac{23816438986321813016005}{39594094982337774529605533696} a^{13} + \frac{4289436119996287379894655}{257361617385195534442435969024} a^{11} + \frac{6934165781315032285266113}{32170202173149441805304496128} a^{9} - \frac{158240179247914498642015}{5948632058644497375241216} a^{7} + \frac{11682257292349953099622033}{251329704477730014103941376} a^{5} - \frac{854601824624159547736521}{3927026632464531470374084} a^{3} - \frac{123376482502820956006603}{604157943456081764672936} a$

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

Class group and class number

$C_{2}\times C_{36}\times C_{36763092}$, which has order $2646942624$ (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:  \( 1907649489.937839 \) (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

\(\Q(\sqrt{-38}) \), 3.3.29241.1, 6.0.8317790995968.2, 9.9.532962204162830310969.10

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 $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$2$2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.9.22.7$x^{9} + 18 x^{8} + 9 x^{7} + 21 x^{6} + 18 x^{5} + 42$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.7$x^{9} + 18 x^{8} + 9 x^{7} + 21 x^{6} + 18 x^{5} + 42$$9$$1$$22$$C_9$$[2, 3]$
19Data not computed