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Properties

Label 18.0.72436308017...1552.4
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 $1916449632$ (GRH)
Class group $[2, 36, 26617356]$ (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![8032970866688, 0, 17155059351552, 0, 10319094448128, 0, 1283370835968, 0, 66282292224, 0, 1730990592, 0, 24322584, 0, 182628, 0, 684, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 684*x^16 + 182628*x^14 + 24322584*x^12 + 1730990592*x^10 + 66282292224*x^8 + 1283370835968*x^6 + 10319094448128*x^4 + 17155059351552*x^2 + 8032970866688)
 
gp: K = bnfinit(x^18 + 684*x^16 + 182628*x^14 + 24322584*x^12 + 1730990592*x^10 + 66282292224*x^8 + 1283370835968*x^6 + 10319094448128*x^4 + 17155059351552*x^2 + 8032970866688, 1)
 

Normalized defining polynomial

\( x^{18} + 684 x^{16} + 182628 x^{14} + 24322584 x^{12} + 1730990592 x^{10} + 66282292224 x^{8} + 1283370835968 x^{6} + 10319094448128 x^{4} + 17155059351552 x^{2} + 8032970866688 \)

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$$\chi_{4104}(1,·)$, $\chi_{4104}(901,·)$, $\chi_{4104}(205,·)$, $\chi_{4104}(1681,·)$, $\chi_{4104}(2005,·)$, $\chi_{4104}(985,·)$, $\chi_{4104}(25,·)$, $\chi_{4104}(829,·)$, $\chi_{4104}(3973,·)$, $\chi_{4104}(2209,·)$, $\chi_{4104}(625,·)$, $\chi_{4104}(745,·)$, $\chi_{4104}(877,·)$, $\chi_{4104}(1021,·)$, $\chi_{4104}(3313,·)$, $\chi_{4104}(2293,·)$, $\chi_{4104}(1405,·)$, $\chi_{4104}(1873,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{16} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{128} a^{9} - \frac{1}{32} a^{7} + \frac{1}{32} a^{5} + \frac{3}{16} a^{3}$, $\frac{1}{512} a^{10} - \frac{1}{128} a^{8} - \frac{7}{128} a^{6} - \frac{5}{64} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{1024} a^{11} - \frac{1}{256} a^{9} - \frac{7}{256} a^{7} + \frac{11}{128} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{16384} a^{12} - \frac{1}{4096} a^{10} - \frac{7}{4096} a^{8} + \frac{11}{2048} a^{6} + \frac{9}{128} a^{4} - \frac{1}{32} a^{2} + \frac{1}{4}$, $\frac{1}{32768} a^{13} - \frac{1}{8192} a^{11} - \frac{7}{8192} a^{9} + \frac{11}{4096} a^{7} - \frac{23}{256} a^{5} - \frac{1}{64} a^{3} - \frac{3}{8} a$, $\frac{1}{22675456} a^{14} - \frac{27}{5668864} a^{12} + \frac{3233}{5668864} a^{10} - \frac{9225}{2834432} a^{8} - \frac{12845}{354304} a^{6} - \frac{111}{44288} a^{4} + \frac{443}{2768} a^{2} + \frac{99}{692}$, $\frac{1}{20362559488} a^{15} - \frac{61269}{5090639872} a^{13} - \frac{2110135}{5090639872} a^{11} + \frac{1888931}{2545319936} a^{9} - \frac{2205631}{79541248} a^{7} - \frac{4362133}{39770624} a^{5} + \frac{607943}{4971328} a^{3} - \frac{76503}{310708} a$, $\frac{1}{36248827338585617863595765149990912} a^{16} - \frac{83814519871388614363362127}{9062206834646404465898941287497728} a^{14} - \frac{18692694484137321268604228911}{9062206834646404465898941287497728} a^{12} - \frac{2747274059141192748643725932785}{4531103417323202232949470643748864} a^{10} + \frac{2276124769638794616973532810829}{566387927165400279118683830468608} a^{8} + \frac{1526840969153677882779953459263}{70798490895675034889835478808576} a^{6} - \frac{29491946225965570120532833095}{1106226420244922420153679356384} a^{4} - \frac{22952250939477710428748597837}{1106226420244922420153679356384} a^{2} - \frac{35058116072949339394254747}{153984746693335526190656926}$, $\frac{1}{72497654677171235727191530299981824} a^{17} - \frac{146507217206305334912699}{18124413669292808931797882574995456} a^{15} - \frac{58525789189456115881808152335}{18124413669292808931797882574995456} a^{13} + \frac{1462255915902775637500438673687}{9062206834646404465898941287497728} a^{11} - \frac{3190366671128533634554724792265}{1132775854330800558237367660937216} a^{9} - \frac{71984483578874072274965179785}{8849811361959379361229434851072} a^{7} + \frac{833856424204912723315728935}{1106226420244922420153679356384} a^{5} - \frac{34019091847284725375930485627}{1106226420244922420153679356384} a^{3} - \frac{36964668275459952866048573995}{276556605061230605038419839096} a$

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

Class group and class number

$C_{2}\times C_{36}\times C_{26617356}$, which has order $1916449632$ (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:  \( 780956274.3587214 \) (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, 9.9.532962204162830310969.7

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 $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$$[3]^{3}$
$3$3.9.22.3$x^{9} + 9 x^{7} + 12 x^{6} + 18 x^{5} + 69$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.3$x^{9} + 9 x^{7} + 12 x^{6} + 18 x^{5} + 69$$9$$1$$22$$C_9$$[2, 3]$
19Data not computed